Functions Reference
Logarithmic Functions
A logarithm is an exponent question. Logarithmic functions undo exponential functions, compress large scales, and make it possible to solve equations where the variable is in an exponent.
Fact Table
| Situation | Use | What to Watch |
|---|---|---|
| Rewrite a log or exponential statement. | $\log_b(x)=y$ means $b^y=x$. | The base stays the base. |
| Evaluate a logarithm by hand. | Ask what exponent makes the base produce the argument. | The argument must be positive. |
| Expand or condense logs. | Product, quotient, and power rules. | Move coefficients to exponents before condensing. |
| Use a calculator for an unusual base. | Change of base. | Argument goes on top; base goes on bottom. |
| Solve a log equation. | Condense to one log, rewrite exponentially. | Check for extraneous answers from domain restrictions. |
| Compare values on a log scale. | Subtract log-scale values, then use the base. | A difference of 1 on a base-10 scale means a factor of 10. |
Content Formulas
Meaning of a Log
$$\log_b(x)=y\quad \Longleftrightarrow \quad b^y=x$$
Log Rules
$$\log_b(MN)=\log_b M+\log_b N$$ $$\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N$$ $$\log_b(M^p)=p\log_b M$$
Change of Base
$$\log_b(x)=\frac{\log x}{\log b}=\frac{\ln x}{\ln b}$$
Inverse Relationship
$$\log_b(b^x)=x$$ $$b^{\log_b x}=x$$
For real-valued logarithms, the base must be positive and not equal to 1, and the argument must be positive.
Classic Examples
Translate a Log
Rewrite $\log_2(32)=5$ in exponential form.
Solution MoveKeep the base as the exponential base. The log value becomes the exponent.
$$\log_2(32)=5$$
$$2^5=32$$
Evaluate a Log
Evaluate $\log_3(81)$.
Solution MoveAsk what exponent on 3 produces 81.
$$3^x=81$$
$$3^x=3^4$$
$$x=4$$
$$\log_3(81)=4$$
Expand a Log
Expand $\log_2(8x^3)$.
Solution MoveProducts become sums, and powers become coefficients.
$$\log_2(8x^3)=\log_2 8+\log_2(x^3)$$
$$\log_2(8x^3)=3+3\log_2 x$$
Condense Logs
Condense $2\log x+\log 3$.
Solution MoveMove coefficients to exponents first, then combine addition as multiplication.
$$2\log x+\log 3=\log(x^2)+\log 3$$
$$2\log x+\log 3=\log(3x^2)$$
Change of Base
Evaluate $\log_5(80)$.
Solution MovePut the log of the argument over the log of the base.
$$\log_5(80)=\frac{\log 80}{\log 5}$$
$$\log_5(80)\approx 2.72$$
Solve a Log Equation
Solve $\log_2(x-1)+\log_2(x+1)=3$.
Solution MoveCondense to one logarithm, rewrite exponentially, then check the domain.
$$\log_2((x-1)(x+1))=3$$
$$(x-1)(x+1)=2^3$$
$$x^2-1=8$$
$$x^2=9$$
$$x=3,\quad x=-3$$
Check the original logarithms. The arguments $x-1$ and $x+1$ must both be positive.
$$x=3$$
Solve an Exponential Equation
Solve $5e^{0.4t}=30$.
Solution MoveIsolate the exponential expression, then take the natural log.
$$e^{0.4t}=6$$
$$0.4t=\ln 6$$
$$t=\frac{\ln 6}{0.4}$$
$$t\approx 4.48$$
Log Scale Comparison
Compare earthquakes of magnitude 6.4 and 4.4 on a base-10 amplitude scale.
Solution MoveSubtract the scale values, then raise 10 to that difference.
$$6.4-4.4=2$$
$$10^2=100$$