Logarithms
A video you posted gets twice as many views every day, starting from 1 view. When does it reach 8 views? You can solve $2^x = 8$ in your head: $x = 3$ days. But when does it reach 10 views? No whole number works, because 10 is between $2^3 = 8$ and $2^4 = 16$. You need an operation that asks, "2 to what power gives 10?" That operation is the logarithm. It shows up wherever things grow or shrink by multiplying: savings accounts, earthquakes, sound, acidity, and radioactive decay.
What is a logarithm?
Two bases get special names. The common logarithm has base 10, and you write it without the little base: $\log x = \log_{10} x$. The natural logarithm has base $e$ and is written $\ln x$. You will meet $\ln$ later on this page.
Calculate the value of the following logarithms by directly applying the definition: $\log_2 64$ $\log_3 \frac{1}{9}$ $\log_{10} 0,001$ $\log_5 \sqrt{5}$
Calculate the value of the following logarithms by directly applying the definition:
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$\log_2 64$
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$\log_3 \frac{1}{9}$
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$\log_{10} 0,001$
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$\log_5 \sqrt{5}$
Find the value of the unknown $x$ in each case using the definition of a logarithm: $\log_x 81 = 4$ $\log_2 x = 5$ $\log_7 x = -2$
Find the value of the unknown $x$ in each case using the definition of a logarithm:
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$\log_x 81 = 4$
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$\log_2 x = 5$
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$\log_7 x = -2$
The graph of a logarithm
Since $\log_b a = c$ means $b^c = a$, the functions $y = \log_b x$ and $y = b^x$ are inverse functions. Swapping the inputs and outputs reflects the graph over the line $y = x$.
The point $(0, 1)$ on the exponential becomes $(1, 0)$ on the logarithm, and $(1, 2)$ becomes $(2, 1)$. Every point flips its coordinates.
Try it with the slider below. A bigger base makes $y = b^x$ climb faster, and its inverse log climb more slowly.
Reading a logarithmic scale
Some graphs have a logarithmic vertical axis: the tick marks are $10^1, 10^2, 10^3, \dots$, evenly spaced, so each step up multiplies by 10. On such a graph, a straight line means exponential growth.
The following graph represents the population growth of two colonies of bacteria, A and B, in a laboratory. The scale of the vertical axis (which indicates the number of…
The following graph represents the population growth of two colonies of bacteria, A and B, in a laboratory. The scale of the vertical axis (which indicates the number of bacteria $N(t)$) is logarithmic in base 10, while the horizontal axis (time in hours) is on a linear scale. Observing the line that describes the behavior of colony A, find its associated exponential function $N(t) = N_0 \cdot 10^{k \cdot t}$, knowing that it passes exactly through the points $(0, 10^2)$ and $(4, 10^4)$. At what instant $t$ will colony A reach exactly one million bacteria ($10^6$)?
Properties of logarithms
Logs are exponents, and exponent rules turn multiplication into addition. Notice that $\log(100 \cdot 1000) = \log(100{,}000) = 5$, and also $\log 100 + \log 1000 = 2 + 3 = 5$. That is no coincidence.
You can also use $\log 10 = 1$ to build numbers like 5, 25, or 0.02 from a single known log.
Knowing that $\log 2 \approx 0,301$ and $\log 3 \approx 0,477$, calculate the approximate numerical value of $\log 6$ and $\log 1,5$ using the properties of logarithms.
Knowing that $\log 2 \approx 0,301$ and $\log 3 \approx 0,477$, calculate the approximate numerical value of $\log 6$ and $\log 1,5$ using the properties of logarithms.
Reduce the following mathematical expression to a single logarithm and calculate its numerical value: $\log 5 + \log 20 - \log 10$
Reduce the following mathematical expression to a single logarithm and calculate its numerical value:
$\log 5 + \log 20 - \log 10$
Knowing exclusively that $\log 2 \approx 0,301$, calculate the approximate value of $\log 25$ and $\log \sqrt[3]{0,02}$ without entering the direct operation into the…
Knowing exclusively that $\log 2 \approx 0,301$, calculate the approximate value of $\log 25$ and $\log \sqrt[3]{0,02}$ without entering the direct operation into the calculator, relying only on the properties of logarithms.
Expanding and condensing logarithms
The three rules work in both directions.
| Direction | What you do | Example |
|---|---|---|
| Expand | Split one log into several | $\log\frac{x^2}{y} = 2\log x - \log y$ |
| Condense | Combine several logs into one | $\log x + \log y = \log(xy)$ |
Apply the properties of logarithms in reverse to express the following combination as a single logarithm: $3 \log x + \frac{1}{2} \log y$
Apply the properties of logarithms in reverse to express the following combination as a single logarithm:
$3 \log x + \frac{1}{2} \log y$
Expand the following logarithmic expression by applying all possible properties, so that it is expressed in terms of $\log x$, $\log y$ and $\log z$:…
Expand the following logarithmic expression by applying all possible properties, so that it is expressed in terms of $\log x$, $\log y$ and $\log z$:
$\log \left( \frac{x^3 \cdot \sqrt{y}}{z^2} \right)$
Group the following algebraic expression into a single logarithm, applying the corresponding properties: $2 \log a - 3 \log b + \log c - \frac{1}{3} \log d$
Group the following algebraic expression into a single logarithm, applying the corresponding properties:
$2 \log a - 3 \log b + \log c - \frac{1}{3} \log d$
Change of base
Your calculator has only two log keys: $\log$ (base 10) and $\ln$ (base $e$). To evaluate a log in any other base, you convert it.
A special case is worth knowing. If you change $\log_a b$ to base $b$, you get $\log_a b = \dfrac{\log_b b}{\log_b a} = \dfrac{1}{\log_b a}$.
Use the change of base formula to calculate the value of $\log_5 120$ using only the common logarithm ($\log$) or natural logarithm ($\ln$) key on your calculator. Round…
Use the change of base formula to calculate the value of $\log_5 120$ using only the common logarithm ($\log$) or natural logarithm ($\ln$) key on your calculator. Round the result to three decimal places.
Simplify the following combined logarithmic expression and calculate its exact numerical value without using a calculator:…
Simplify the following combined logarithmic expression and calculate its exact numerical value without using a calculator:
$\frac{\log_2 8 + \log_3 81 - \log 1000}{\log_5 125 \cdot \log_4 \sqrt{4}}$
Prove, using the change of base formula and the fundamental properties of logarithms, that $\log_a b \cdot \log_b a = 1$ for any numbers $a, b > 0$ and different from…
Prove, using the change of base formula and the fundamental properties of logarithms, that $\log_a b \cdot \log_b a = 1$ for any numbers $a, b > 0$ and different from $1$. Once proven, apply this result to mentally calculate the value of $\log_8 2$ knowing beforehand that $\log_2 8 = 3$.
Solving exponential equations
If the unknown is in an exponent and both sides cannot be written with the same base, take a logarithm of both sides. The power rule brings the exponent down where you can reach it.
Real situations lead to these equations. In exponential models, the quantity at time $t$ is the starting amount times a growth factor to the power $t$.
Calculate the value of $x$ in the exponential equation $3^x = 45$. Express the final result rounded to two decimal places (you may use a calculator for the final step).
Calculate the value of $x$ in the exponential equation $3^x = 45$. Express the final result rounded to two decimal places (you may use a calculator for the final step).
A population of bacteria doubles every hour that passes. If initially there are $1500$ bacteria, the number of bacteria after $t$ hours is given by the function…
A population of bacteria doubles every hour that passes. If initially there are $1500$ bacteria, the number of bacteria after $t$ hours is given by the function $P(t) = 1500 \cdot 2^t$. Exactly how much time must pass for the population to reach the figure of $100000$ bacteria?
Solve the exponential equation $5^{x+1} = 2^{3x-2}$ by taking common logarithms on both sides and solving for the unknown $x$. Express the final result rounded to three…
Solve the exponential equation $5^{x+1} = 2^{3x-2}$ by taking common logarithms on both sides and solving for the unknown $x$. Express the final result rounded to three decimal places.
Solving logarithmic equations
If the unknown is inside a logarithm, undo the log by rewriting the equation in exponential form. If there are several logs, condense them into one first.
For a system, treat $\log x$ and $\log y$ as the unknowns and use elimination.
Solve the logarithmic equation $\log_3 (2x + 5) = 2$ by applying the definition of a logarithm to solve for the unknown $x$.
Solve the logarithmic equation $\log_3 (2x + 5) = 2$ by applying the definition of a logarithm to solve for the unknown $x$.
Solve the following system of logarithmic equations, finding the values of $x$ and $y$: $$ \begin{cases} \log x + \log y = 3 \\ \log x - \log y = 1 \end{cases} $$
Solve the following system of logarithmic equations, finding the values of $x$ and $y$:
$$ \begin{cases} \log x + \log y = 3 \\ \log x - \log y = 1 \end{cases} $$Growth and decay models
Money in a savings account, a phone that loses value, and followers on a viral account all follow $y = a \cdot b^t$. The base $b$ is the growth factor per time period. If something grows $r\%$ per period, $b = 1 + r$ (as a decimal). If it shrinks $r\%$, $b = 1 - r$.
For compound interest once a year, $A = P(1 + r)^t$, where $P$ is the principal. More generally, $A = P\left(1 + \tfrac{r}{n}\right)^{nt}$ for $n$ compoundings per year.
A capital of $\$5000$ is deposited in a bank at a compound interest rate of $3,5\,\%…
A capital of $\$5000$ is deposited in a bank at a compound interest rate of $3,5\,\%$ per year. How many full years must pass for the accumulated capital to exceed $\$7000$?
Remember that the formula for the final capital is $C_f = C_i \cdot \left(1 + \frac{r}{100}\right)^t$.
A new vehicle has an initial cost of $\$24000$. Due to market depreciation, its value depreciates by $12\,\%…
A new vehicle has an initial cost of $\$24000$. Due to market depreciation, its value depreciates by $12\,\%$ each year. The function that describes its residual value after $t$ years is $V(t) = 24000 \cdot (0,88)^t$. How many years must pass for the value of the car to be reduced to exactly half of its original price?
On a well-known social network, the number of followers of a viral content account grows following an exponential model $S(t) = S_0 \cdot k^t$, where $t$ is measured in…
On a well-known social network, the number of followers of a viral content account grows following an exponential model $S(t) = S_0 \cdot k^t$, where $t$ is measured in days. It is known that on the initial day ($t = 0$) the account had $200$ followers and on day $5$ it reached $6400$ followers. First, calculate the exact value of the growth constant $k$. Then, determine on exactly which day it will surpass the one million followers mark.
Natural logarithms and base $e$
Science and calculus use a special base: the irrational number $e \approx 2.71828$. Its log is the natural logarithm, $\ln x = \log_e x$. Everything you know about logs still works, with base $e$.
Continuous growth and decay use $N(t) = N_0 e^{kt}$, with $k > 0$ for growth and $k < 0$ for decay. To get $t$ out of the exponent, take $\ln$ of both sides.
The amount of Carbon-14 ($C-14$) in an organic archaeological remain follows the law of radioactive decay $N(t) = N_0 \cdot e^{-\lambda t}$, where the decay constant is…
The amount of Carbon-14 ($C-14$) in an organic archaeological remain follows the law of radioactive decay $N(t) = N_0 \cdot e^{-\lambda t}$, where the decay constant is $\lambda \approx 0,0001209$ and $t$ is the time elapsed in years. If laboratory analysis detects that only $30\,\%$ of its original amount of $C-14$ remains in a prehistoric bone fragment (i.e., $N(t) = 0,30 \cdot N_0$), what is the estimated age of the bone?
Logarithmic scales in the real world
Some quantities range from tiny to gigantic. A logarithmic scale squeezes them into small numbers: each step up on the scale multiplies the real quantity by a fixed factor, usually 10.
The pH of a chemical solution is defined by the formula $pH = -\log [H^+]$, where $[H^+]$ represents the concentration of hydrogen ions measured in mol/L. If a sample of…
The pH of a chemical solution is defined by the formula $pH = -\log [H^+]$, where $[H^+]$ represents the concentration of hydrogen ions measured in mol/L. If a sample of lemon juice has a hydrogen ion concentration of $10^{-3}$ mol/L, what is its pH value?
The magnitude of an earthquake on the Richter scale is calculated using the formula $M = \log \left( \frac{A}{A_0} \right)$, where $A$ is the maximum amplitude of the…
The magnitude of an earthquake on the Richter scale is calculated using the formula $M = \log \left( \frac{A}{A_0} \right)$, where $A$ is the maximum amplitude of the seismic waves and $A_0$ is a standard reference amplitude. If a specific earthquake registers an amplitude $1000$ times greater than $A_0$, what is its magnitude on the Richter scale? Furthermore, determine how many times greater the amplitude of an earthquake of magnitude $6$ is compared to one of magnitude $4$.
The sound intensity level, measured in decibels (dB), is calculated with the physical formula $L = 10 \cdot \log \left( \frac{I}{I_0} \right)$, where $I$ is the sound…
The sound intensity level, measured in decibels (dB), is calculated with the physical formula $L = 10 \cdot \log \left( \frac{I}{I_0} \right)$, where $I$ is the sound intensity in $\text{W/m}^2$ and $I_0 = 10^{-12} \text{ W/m}^2$ represents the minimum threshold of human hearing. If the noise of heavy traffic on an avenue registers a level of $85$ dB, what is its acoustic intensity $I$? If a construction drill emits with double the acoustic intensity of that traffic, exactly how many decibels will the sound level meter register?
Back to the start: your video doubles its views every day, so you need $2^x = 10$. Take logs: $x\log 2 = \log 10 = 1$, so $x = \dfrac{1}{\log 2} \approx \dfrac{1}{0.301} \approx 3.32$ days. That is between 3 and 4, just as we predicted.