Before you start, make sure you know: Translations (Integrated Math I)
Maya puts $\$500$ into a savings account that earns 6% interest every year. After how many years will she have $\$1{,}000$? You can write the equation $500(1.06)^t = 1000$, but how do you get $t$ out of the exponent? You need an operation that undoes powers. That operation is the logarithm, and in this page you will learn how to calculate it, graph it, and use it to solve equations.
What a Logarithm Means
A logarithm answers one question: "What exponent do I need?" Since $2^5 = 32$, the exponent that turns $2$ into $32$ is $5$. We write that as $\log_2 32 = 5$.
Two bases are so common that they get their own keys on a calculator. The common logarithm is base 10: $\log x$ means $\log_{10} x$. The natural logarithm is base $e \approx 2.718$: $\ln x$ means $\log_e x$.
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Calculate the value of the following logarithms by directly applying the definition, without using a calculator: a) $\log_2 32$ b) $\log_3 \frac{1}{9}$ c) $\log 10000$
Calculate the value of the following logarithms by directly applying the definition, without using a calculator:
Find the value of the unknown $x$ in each of the following equalities using the definition of a logarithm: a) $\log_x 25 = 2$ b) $\log_5 x = 3$ c) $\log_2 64 = x$
Find the value of the unknown $x$ in each of the following equalities using the definition of a logarithm:
In chemistry, the acidity level of a solution is measured by its pH, which is defined using a base 10 logarithm with the formula $pH = -\log[H^+]$, where $[H^+]$ is the…
In chemistry, the acidity level of a solution is measured by its pH, which is defined using a base 10 logarithm with the formula $pH = -\log[H^+]$, where $[H^+]$ is the concentration of hydrogen ions in moles per liter. If a sample of lemon juice has a hydrogen ion concentration of $10^{-3}$ mol/L, calculate its pH.
Move the slider and watch the graph. It never touches the $y$-axis, and it never goes to the left of it.
To find the domain of a more complicated log function, set the argument greater than $0$ and solve the inequality. The output of a log can be any real number, but the argument must be positive.
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Determine the domain of definition of the following logarithmic functions: a) $f(x) = \log(x - 5)$ b) $g(x) = \ln(2x + 8)$
Determine the domain of definition of the following logarithmic functions:
Notice that the two graphs are mirror images across the $x$-axis. In fact $\log_{1/2} x = -\log_2 x$.
The point $(a, 1)$ is a quick way to read the base from a graph: find where the curve is at height $1$, and the $x$-coordinate of that point is the base. For example, if a graph passes through $(5, 1)$, the function is $\log_5 x$. Also, $\ln x$ is increasing because $e \approx 2.718 > 1$.
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Indicate, reasoning your answer and without needing to draw them, which of the following functions are increasing and which are decreasing: a) $f(x) = \log_3 x$ b)…
Indicate, reasoning your answer and without needing to draw them, which of the following functions are increasing and which are decreasing:
Observe the graphs of the logarithmic functions represented in the image above. Match, justifying your answer, Function A and Function B with their respective analytical…
Observe the graphs of the logarithmic functions represented in the image above. Match, justifying your answer, Function A and Function B with their respective analytical expressions:
The exponential $y = a^x$ turns an exponent into a number. The logarithm $y = \log_a x$ does the opposite, turning a number back into an exponent. They are inverse functions.
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The image shows the graph of the exponential function $f(x) = e^x$ and its inverse function $g(x)$. a) Write the analytical expression of $g(x)$. b) Explain the…
The image shows the graph of the exponential function $f(x) = e^x$ and its inverse function $g(x)$.
a) Write the analytical expression of $g(x)$.
b) Explain the geometric meaning of the dotted orange line $y = x$ in relation to both curves.
c) Determine the domain and range of both functions, justifying how they are interchanged between a function and its inverse.
Logarithmic graphs move just like any other graph you have translated.
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Given the function $f(x) = \log_3(12 - 3x)$: a) Determine its domain analytically. b) Indicate the equation of its vertical asymptote. c) In which direction (left or…
Given the function $f(x) = \log_3(12 - 3x)$:
a) Determine its domain analytically.
b) Indicate the equation of its vertical asymptote.
c) In which direction (left or right) does the graph of this function open with respect to its asymptote?
From the graph of the function $f(x)$ shown above, which corresponds to a translated logarithm of the form $f(x) = \log_a (x + b)$: a) Indicate the value of $b$ by…
From the graph of the function $f(x)$ shown above, which corresponds to a translated logarithm of the form $f(x) = \log_a (x + b)$:
a) Indicate the value of $b$ by observing the vertical asymptote.
b) Use the point $B(1, 2)$ to calculate the base $a$ of the logarithm.
c) Write the complete analytical expression of the function.
It is known that a logarithmic function of the form $f(x) = \log_a(x + b)$ passes through the origin of coordinates $(0, 0)$ and through the point $(6, 1)$. Set up a…
It is known that a logarithmic function of the form $f(x) = \log_a(x + b)$ passes through the origin of coordinates $(0, 0)$ and through the point $(6, 1)$.
Set up a system of equations to find the values of the parameters $a$ and $b$. Once found, write the final analytical expression of the function and indicate its vertical asymptote.
Given that $\log 2 = 0,301$ and $\log 3 = 0,477$, calculate the value of $\log 12$ and $\log 5$ using the properties of logarithms, without directly using the log key on…
Given that $\log 2 = 0,301$ and $\log 3 = 0,477$, calculate the value of $\log 12$ and $\log 5$ using the properties of logarithms, without directly using the log key on the calculator for those numbers. (Hint: remember that $5 = \frac{10}{2}$).
Expanding means using the properties to write one log as a sum and difference of simple logs. Condensing is the reverse: combine several logs into a single one.
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Expand the following expression by applying the properties of logarithms (logarithm of a product, quotient and power), so that only logarithms of simple variables…
Expand the following expression by applying the properties of logarithms (logarithm of a product, quotient and power), so that only logarithms of simple variables remain:
When the unknown is in an exponent, take a logarithm of both sides. The power rule pulls the exponent down so you can solve for it.
Here is the graph of Maya's account ($y = 1.06^t$) and the line $y = 2$, which means the balance has doubled. They meet at about $t = 11.9$.
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A capital is invested in a bank fund that offers an annual compound interest. The formula governing the accumulated capital $C$ over the years $t$ is…
A capital is invested in a bank fund that offers an annual compound interest. The formula governing the accumulated capital $C$ over the years $t$ is $C = C_0 \cdot 1,04^t$, where $C_0$ is the initial capital. If $2000$ dollars are invested, determine how many full years must pass for the capital to exceed $3000$ dollars. Use logarithms to solve for the unknown.
Carbon-14 dating is used to date ancient organic remains. The proportion of C-14 decreases exponentially with time according to the function…
Carbon-14 dating is used to date ancient organic remains. The proportion of C-14 decreases exponentially with time according to the function $P(t) = 100 \cdot 0,5^{\frac{t}{5730}}$, where $P$ is the percentage of C-14 remaining compared to the original and $t$ is the time elapsed in years.
If a fossil is discovered that retains only $15\%$ of its original Carbon-14, what is the estimated age of the fossil? Express the result in years and round to the nearest whole number.
Some quantities, like sound intensity or earthquake energy, range from tiny to enormous. A logarithm squeezes that huge range into small, easy numbers.
Look at the graph: $x$ goes from $1$ all the way to $100$, but $\log x$ only goes from $0$ to $2$. A logarithm grows very slowly.
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Sound intensity is measured in decibels (dB) using a logarithmic function. The formula is $d = 10 \cdot \log\left(\frac{I}{I_0}\right)$, where $d$ is the decibels, $I$…
Sound intensity is measured in decibels (dB) using a logarithmic function. The formula is $d = 10 \cdot \log\left(\frac{I}{I_0}\right)$, where $d$ is the decibels, $I$ is the sound intensity and $I_0$ is the threshold intensity of hearing (a constant). Make the intensity $I$ the subject in terms of the decibels $d$ and $I_0$.
The Richter scale for measuring the magnitude of earthquakes is logarithmic and is defined as $M = \log\left(\frac{A}{A_0}\right)$, where $A$ is the amplitude of the…
The Richter scale for measuring the magnitude of earthquakes is logarithmic and is defined as $M = \log\left(\frac{A}{A_0}\right)$, where $A$ is the amplitude of the seismic wave and $A_0$ is a reference amplitude.
An earthquake in city Alpha had a magnitude of $6$ on the Richter scale, and another in city Beta had a magnitude of $8$. How many times greater was the amplitude of the seismic wave of the Beta earthquake compared to that of Alpha? Prove your reasoning mathematically.
In computer science, the execution time $T$ of a certain sorting algorithm depends on the amount of data $n$ to be processed, and follows a function of the type…
In computer science, the execution time $T$ of a certain sorting algorithm depends on the amount of data $n$ to be processed, and follows a function of the type $T(n) = k \cdot n \cdot \log_2 n$, where $k$ is a constant specific to the computer.
If when processing $n = 8$ data items the computer takes $T = 24$ milliseconds:
a) Calculate the value of the constant $k$.
b) How long will the same computer take to process $n = 32$ data items?