Logarithms
Your new channel's follower count doubles every week. You can see that $2^x = 8$ takes $x = 3$ weeks. But how many weeks until you have 10 times as many followers? That means solving $2^x = 10$. No whole number works, because $10$ sits between $2^3 = 8$ and $2^4 = 16$. You need an operation that answers "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?
A power starts with the exponent and gives you the number. A logarithm goes the other way: it starts with the number and gives you the exponent.
Two bases are so common that they get their own calculator keys. The common logarithm $\log x$ means $\log_{10} x$. The natural logarithm $\ln x$ means $\log_e x$, where $e \approx 2.718$. 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:
-
$\log_2 64$
-
$\log_3 \frac{1}{9}$
-
$\log_{10} 0,001$
-
$\log_5 \sqrt{5}$
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:
-
$\log_x 81 = 4$
-
$\log_2 x = 5$
-
$\log_7 x = -2$
The graph: the inverse of an exponential
Since $\log_a b = c$ means the same as $a^c = b$, the functions $y = \log_a x$ and $y = a^x$ are inverse functions. Swap the $x$ and $y$ coordinates of any point on one graph and you get a point on the other. So the two graphs are mirror images across the line $y = x$.
Properties of logarithms
Logarithms are exponents, so the rules of exponents turn into rules for logs. When you multiply powers with the same base you add the exponents. That is why multiplying inside a log becomes adding outside it.
Roots are powers too, since $\sqrt{x} = x^{1/2}$ and $\sqrt[3]{x} = x^{1/3}$. So $\log\sqrt{x} = \tfrac12\log x$ and $\log\sqrt[3]{x} = \tfrac13\log x$.
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 evaluating the direct operation on a…
Knowing exclusively that $\log 2 \approx 0,301$, calculate the approximate value of $\log 25$ and $\log \sqrt[3]{0,02}$ without evaluating the direct operation on a calculator, relying only on the properties of logarithms.
Expanding and condensing logarithms
Use the properties left to right to expand a single log into a sum of simple logs. Use them right to left to condense several logs into one.
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 change the base.
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 $\log_2 8 = 3$.
Exponential equations
When the unknown is in the exponent and both sides cannot be written with the same base, take a logarithm of both sides. The power rule then brings the exponent down where you can solve for it.
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).
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.
Logarithmic equations
When the unknown is inside a logarithm, use the definition to undo the log. Then solve the equation you get. Finally, always check that every log in the original equation has a positive argument.
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 problems
Growth or decay by a fixed percent each period follows $y = a\cdot b^t$. A percent increase of $r$ (as a decimal) gives growth factor $b = 1 + r$. A percent decrease gives $b = 1 - r$. Asking when something reaches a target means solving for $t$, which is an exponent, so you need a logarithm.
For compound interest, the amount after $t$ years is $A = P(1 + r)^t$. Some problems write the rate as a percent and use $\left(1 + \frac{r}{100}\right)^t$. It is the same formula.
A principal of $\$5000$ is deposited in a bank at a compound interest rate of $3,5\,\%…
A principal 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 amount to exceed $\$7000$?
Remember that the formula for the compound amount is $C_f = C_i \cdot \left(1 + \frac{r}{100}\right)^t$.
A population of bacteria doubles every hour. 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. 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 $100000$ bacteria?
A new vehicle has an initial cost of $\$24000$. Due to market depreciation, its value decreases by $12\,\%…
A new vehicle has an initial cost of $\$24000$. Due to market depreciation, its value decreases 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?
The natural logarithm and the number $e$
Many natural processes, such as radioactive decay and continuous growth, are written with the base $e \approx 2.718$. To undo $e^{\text{something}}$, use the natural logarithm $\ln x = \log_e x$. It follows every rule above.
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?
On a popular 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 popular 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.
Logarithmic scales: pH and earthquakes
Some quantities span such a huge range that a regular number line is useless. A logarithmic scale fixes that: each step up means "times $10$" instead of "plus a fixed amount."
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 $4$.
Decibels and straight-line log graphs
Sound intensity covers an enormous range, so loudness is measured in decibels with a log.
Now a different kind of log picture. On a semi-log graph, the vertical axis is logarithmic. Each gridline is $10$ times the one below it, and the horizontal axis is ordinary. An exponential function $N = N_0\cdot10^{kt}$ becomes a straight line there, because $\log N = \log N_0 + kt$ is linear in $t$.
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 double the acoustic intensity of that traffic, exactly how many decibels will the sound level meter register?
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 time $t$ will colony A reach exactly one million bacteria ($10^6$)?
Back to the start: Followers double every week, so we need $2^x = 10$. Take logs: $x\log 2 = \log 10 = 1$, so $x = \dfrac{1}{\log 2} \approx \dfrac{1}{0.301} \approx 3.32$ weeks. That fits between $3$ and $4$, just as $10$ sits between $2^3 = 8$ and $2^4 = 16$.