Exponentials and Logarithms

Precalculus is the course that gets functions ready for calculus. A function is a rule with an input and one output, and here you learn how to graph, change, and reverse them.

Exponentials and Logarithms: the central idea

Exponential functions place the variable in an exponent; logarithms undo that exponent by asking which power of a base produces a positive number.

Words you need

Exponential function

An exponential function is a rule of the form $y=a\cdot b^{x}$ where the variable sits in the exponent, so the output is multiplied by the base $b$ each time $x$ goes up by one.

Logarithm

A logarithm is the exponent you need: $\log_b N$ is the power that turns the base $b$ into $N$, so $\log_{10}1000=3$ because $10^3=1000$.

Base

The base of an exponential or logarithm is the number being multiplied over and over, and it must be positive and not equal to $1$ for the function to be useful.

Natural logarithm

The natural logarithm, written $\ln x$, is the logarithm with base $e\approx2.71828$, the growth rate that appears whenever change is continuous.

Common logarithm

The common logarithm, written $\log x$ with no base shown, means base $10$, which is the convention used in decibels, pH, and the Richter scale.

Half-life and doubling time

Half-life and doubling time are the times it takes a quantity to fall to half or rise to double, and each is found by solving an exponential equation with a logarithm.

What to know before this lesson

Know exponent laws, inverse functions, positive domains, and how repeated multiplication models constant percentage change.

If one of those prerequisites is uncertain, use the Precalculus subject guide to locate the earlier concept before memorizing a procedure.

Exponentials and logarithms: a worked example

Every step, with the arithmetic

1. Step 1 - Start with a solvable equation$5\cdot2^{3x}=320$
2. Step 2 - Get the exponential alone$2^{3x}=320\div5=64$
3. Step 3 - Rewrite the right side with the same base$64=2^6$, so $2^{3x}=2^6$
4. Step 4 - Match the exponents and solve$3x=6$, so $x=2$
5. Step 5 - Check the answer$5\cdot2^{6}=5\cdot64=320$
6. Step 6 - Now try one with no matching base$7^x=50$, and $50$ is not a whole power of $7$
7. Step 7 - Take the log of both sides$x\ln 7=\ln 50$, so $x=\dfrac{\ln 50}{\ln 7}=\dfrac{3.9120}{1.9459}\approx2.0104$
8. Step 8 - Confirm the size is sensible$7^2=49$, which is just under $50$, so an answer a hair above $2$ is exactly right

$2^x=16$ becomes $2^x=2^4$, so $x=4$; equivalently, $x=\log_2 16$.

The logarithm laws, each shown with real numbers

The step-by-step method for exponentials and logarithms

1. Isolate the exponential or logarithmic expression and state domain restrictions.
2. Rewrite both sides with a common base when possible; otherwise apply a logarithm and use its power rule.
3. Solve the resulting equation and reject any value that makes a logarithm input nonpositive.

How to check your answer

Substitute into the original equation and compare exponential growth with a graph or successive-value ratio.

Use intercepts, end behavior, symmetry, and a few exact points to test whether a formula and graph describe the same function.

A mistake that changes the mathematics

Logarithms do not distribute across addition: $\log(a+b)$ is not $\log a+\log b$.

Where you will actually use this

Earthquake magnitude

The Richter scale is a base-$10$ log, so a magnitude $7$ quake shakes the ground $10$ times harder than a magnitude $6$ and $100$ times harder than a magnitude $5$.

Carbon dating

Carbon-14 has a half-life of about $5730$ years, so a sample with $25\%$ left has gone through two half-lives, giving an age near $11{,}460$ years.

Sound levels in decibels

Decibels are logarithmic, so every $10$ dB means ten times the sound power, which is why $90$ dB traffic is far more than merely $50\%$ louder than $60$ dB speech.

How exponentials and logarithms connects to the rest of precalculus

• Inverse functions — Logs and exponentials are the cleanest inverse pair there is, so $\log_b(b^x)=x$ is just the composition check from that lesson.
• Sequences and series — A geometric sequence is an exponential rule evaluated at whole numbers, and a logarithm answers which term first passes a target value.
• Percentages — Constant percentage growth is exactly an exponential: a $6\%$ yearly rise means multiplying by $1.06$ each year, so the base is $1.06$.

Try a transfer problem

Solve $3^{2x-1}=20$ using logarithms and keep enough precision to verify the original equation.

Show the worked answer

Solve $3^{2x-1}=20$. The number $20$ is not a whole power of $3$, so matching bases will not work and you take a logarithm of both sides: $\ln\left(3^{2x-1}\right)=\ln 20$. The power law brings the exponent down in front: $(2x-1)\ln 3=\ln 20$, so $2x-1=\dfrac{\ln 20}{\ln 3}=\dfrac{2.995732}{1.098612}=2.726833$. Add $1$ to get $2x=3.726833$, then halve it: $x\approx1.863416$, or about $1.8634$. The exact form is $x=\dfrac{1}{2}\left(1+\log_3 20\right)$. Keep the extra decimals until the end, because rounding $2x-1$ to $2.73$ too early shifts the answer in the third decimal place. Check it: $2(1.863416)-1=2.726832$, and $3^{2.726832}=e^{2.726832\times1.098612}=e^{2.995731}\approx20.00$. A rough sanity check also works: $3^2=9$ and $3^3=27$, so the exponent $2x-1$ must land between $2$ and $3$, closer to $3$, which puts $x$ between $1.5$ and $2$.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to precalculus practice questions.

Questions about exponentials and logarithms

Why can I never take the log of zero or a negative number?

Because no exponent produces them. $2^x$ gives $8$, $1$, $0.5$, and $0.001$, but it never reaches $0$ and never dips below it, no matter how negative $x$ gets. So $\log_2 0$ and $\log_2(-4)$ have no answer, and the domain of every log is the positive numbers only.

Does $\log(a+b)$ equal $\log a+\log b$?

No. Test it: $\log_2(4+4)=\log_2 8=3$, but $\log_2 4+\log_2 4=2+2=4$. The addition rule works on the other side, since $\log a+\log b=\log(ab)$. Logs turn multiplication into addition, never addition into addition.

What is $e$ and why does it show up everywhere?

$e\approx2.71828$ is the limit of $(1+1/n)^n$ as $n$ grows, which is what compound interest becomes when it is paid continuously instead of yearly. It matters because $e^x$ is the one exponential whose rate of growth equals its own height, which makes it the natural base for anything changing continuously.

My calculator only has log and ln. How do I compute $\log_5 200$?

Use change of base: $\log_5 200=\dfrac{\ln 200}{\ln 5}=\dfrac{5.298317}{1.609438}\approx3.2920$. Either button works as long as you use the same one top and bottom. Check the size: $5^3=125$ and $5^4=625$, so an answer between $3$ and $4$ is right.