The Power Rule and Change of Base for Logarithms

Definition Let $b > 0$ with $b \ne 1$, let $M > 0$, and let $r$ be any real number. Then

To prove it, put $u = \log_b M$, so that $b^u = M$. Then $M^r = \left(b^u\right)^r = b^{ur}$, and by the definition of the logarithm $\log_b\left(M^r\right) = ur = r \log_b M$. Roots are powers too, so for every integer $n \ge 2$ the rule gives

Definition The power rule needs $M > 0$. The left side $\log_b\left(M^r\right)$ can be defined for a negative $M$, for example when $r$ is an even integer, but then $\log_b M$ is not defined. For every real $M \ne 0$ and every integer $k$ the correct statement is

So $\ln\left(x^2\right) = 2 \ln \lvert x \rvert$ for every $x \ne 0$, while $\ln\left(x^2\right) = 2 \ln x$ holds only for $x > 0$.

Definition Let $a$ and $b$ be positive numbers, neither equal to $1$, and let $x > 0$. Then

To prove it, put $y = \log_b x$, so that $b^y = x$. Taking $\log_a$ of both sides and using the power rule gives $y \log_a b = \log_a x$. Since $b \ne 1$, the number $\log_a b$ is not $0$, and dividing by it gives the formula. With the two bases on a calculator,

Taking $x = a$ gives the special case $\log_b a = 1/\log_a b$. Calculus uses the formula to reduce every logarithm to $\ln$, for example in derivatives of exponential and logarithmic functions. The power rule does the same for exponentials. For $b > 0$ and every real $x$ it gives $\ln\left(b^x\right) = x \ln b$, so $b^x = e^{x \ln b}$. Calculus takes this formula as its definition of $b^x$, and the computation shows that it gives the same values as the definition in the lesson on exponential functions.

Definition The base of a logarithm must be a positive real number other than $1$, and in the change of base formula this applies to both bases. The new base $a$ must be valid so that $\log_a x$ and $\log_a b$ exist, and the condition $b \ne 1$ is exactly what keeps the denominator $\log_a b$ away from $0$. When the base contains a variable, as in $\log_x 4$, the expression is defined only for $x > 0$ with $x \ne 1$.

Definition An exact expression names a number with no rounding, using integers, fractions, radicals, constants such as $\pi$ and $e$, and functions such as logarithms. The expressions $\log_3 50$, $\frac{\ln 50}{\ln 3}$ and $\log_3 2 + 2 \log_3 5$ are all exact, and they all name the same number. An exact answer is usually written in simplest form, with every logarithm that equals a rational number replaced by that number, for example $\log_2 8$ by $3$.

Definition A calculator approximation is a decimal that agrees with an exact value to a stated number of decimal places or significant digits. It is written with $\approx$ rather than $=$. Rounding in the middle of a calculation lets the errors grow, so keep full precision in every intermediate step and round only the final answer.