Rational and Logarithmic Functions Reference

Logarithms

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.

Solved Examples

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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.

Log Rules Table

Rule Formula Algebra Meaning
Product rule $\log_b(MN)=\log_b M+\log_b N$ Multiplication inside a log becomes addition outside.
Quotient rule $\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N$ Division inside a log becomes subtraction outside.
Power rule $\log_b(M^p)=p\log_b M$ An exponent inside a log becomes a coefficient outside.
Condensing addition $\log_b M+\log_b N=\log_b(MN)$ Addition of logs becomes multiplication inside one log.
Condensing subtraction $\log_b M-\log_b N=\log_b\left(\frac{M}{N}\right)$ Subtraction of logs becomes division inside one log.
Condensing coefficients $p\log_b M=\log_b(M^p)$ Move coefficients up as exponents before combining logs.

Classic Examples

Translate a Log

Rewrite $\log_2(32)=5$ in exponential form.

Solution Steps
  1. Keep the base as the exponential base.
  2. Use the log value as the exponent and state the equivalent equation: $2^5=32$.
$$\log_2(32)=5$$ $$2^5=32$$

Evaluate a Log

Evaluate $\log_3(81)$.

Solution Steps
  1. Rewrite the logarithm as an exponential equation.
  2. Rewrite $81$ as a power of $3$.
  3. Set the exponents equal.
  4. State the logarithm's value: $\log_3(81)=4$.
$$3^x=81$$ $$3^x=3^4$$ $$x=4$$ $$\log_3(81)=4$$

Expand a Log

Expand $\log_2(8x^3)$.

Solution Steps
  1. Apply the product rule to separate the factors.
  2. Evaluate $\log_2 8$ and apply the power rule.
$$\log_2(8x^3)=\log_2 8+\log_2(x^3)$$ $$\log_2(8x^3)=3+3\log_2 x$$

Condense Logs

Condense $2\log x+\log 3$.

Solution Steps
  1. Move the coefficient to an exponent using the power rule.
  2. Combine addition as multiplication inside one logarithm.
$$2\log x+\log 3=\log(x^2)+\log 3$$ $$2\log x+\log 3=\log(3x^2)$$

Change of Base

Evaluate $\log_5(80)$.

Solution Steps
  1. Put the logarithm of the argument over the logarithm of the base.
  2. Evaluate the quotient and state the approximation: $\log_5(80)\approx2.72$.
$$\log_5(80)=\frac{\log 80}{\log 5}$$ $$\log_5(80)\approx 2.72$$

Solve a Log Equation

Solve $\log_2(x-1)+\log_2(x+1)=3$.

Solution Steps
  1. Condense the logarithms into one logarithm.
  2. Rewrite the logarithmic equation exponentially.
  3. Expand the product and isolate the quadratic terms.
  4. Solve the resulting quadratic equation.
  5. State both candidate solutions.
  6. Check the original logarithm domains and keep $x=3$.
$$\log_2((x-1)(x+1))=3$$ $$(x-1)(x+1)=2^3$$ $$x^2-1=8$$ $$x^2=9$$ $$x=3,\quad x=-3$$ $$x=3$$

Solve an Exponential Equation

Solve $5e^{0.4t}=30$.

Solution Steps
  1. Divide by the coefficient to isolate the exponential.
  2. Take the natural log of both sides.
  3. Divide by the coefficient of $t$.
  4. State the approximate solution: $t\approx4.48$.
$$e^{0.4t}=6$$ $$0.4t=\ln 6$$ $$t=\frac{\ln 6}{0.4}$$ $$t\approx 4.48$$

Log Scale Comparison

Compare earthquakes of magnitude 6.4 and 4.4 on a base-10 amplitude scale.

Solution Steps
  1. Subtract the two base-10 scale values.
  2. Raise $10$ to that difference and state the comparison: $100$ times the amplitude.
$$6.4-4.4=2$$ $$10^2=100$$