Exponential and logarithmic functions: practice problems

Exponent laws and roots

Write everything as powers of the same base. Example: $\sqrt[3]{4} \times \sqrt[3]{16} = 2^{\frac{2}{3}} \times 2^{\frac{4}{3}} = 2^2 = 4$.

To compare numbers, use a common base: if the base is greater than $1$, a larger exponent gives a larger number (the reverse if the base is less than $1$).

Logarithm rules and change of base

Example: $\log_2 3 \cdot \log_3 8 = \frac{\log 3}{\log 2} \cdot \frac{3\log 2}{\log 3} = 3$.

Equations and inequalities

For exponential equations, write both sides as powers of the same base and compare exponents. For $4^x - 6 \cdot 2^x + 8 = 0$, substitute $t = 2^x$ ($t > 0$).

For logarithmic equations and inequalities, first require the arguments to be positive. Example: $\log_2 x + \log_2(x - 2) = 3$ needs $x > 2$; then $x(x - 2) = 8$ gives $x = 4$ ($x = -2$ is rejected).

Common logarithms and digits

A positive integer $N$ has $n$ digits exactly when $10^{n-1} \leq N < 10^n$, i.e. $n - 1 \leq \log_{10} N < n$.

Example: with $\log_{10} 2 = 0.3010$, $\log_{10} 2^{50} = 15.05$, so $2^{50}$ has $16$ digits. Since the fractional part $0.05$ is less than $\log_{10} 2$, the leading digit is $1$.

For $0 < N < 1$: if $-n \leq \log_{10} N < -(n - 1)$, the first non-zero digit is at the $n$-th decimal place.