Definition of the logarithm

The logarithm $\log_a b$ answers one question: $a$ to what power gives $b$? The definition is $\log_a b=c \iff a^c=b$, so $\log_2 8=3$ is just $2^3=8$ rewritten.

Translating the exponent rules into logarithms gives three rules.

In calculus, $\log x$ means the natural logarithm (base $e$). The rule $(\log x)'=\dfrac{1}{x}$ follows from this definition and the derivative of $e^x$.

Examples

1. 01Given $\log_{5}125 = 3$, evaluate:$\displaystyle \log_{5}625 - \log_{5}125$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. $\displaystyle 5^{4}$$\displaystyle = 625$
   2. Subtracting exponents means dividing
   3. $\displaystyle \frac{625}{125}$$\displaystyle = 5^{1}$
   4. In logarithmic form
   5. $\displaystyle \log_{5}\frac{625}{125}$$\displaystyle = 1$
2. 02Write in exponential form:$\displaystyle \log_{2}8 = 3$AnswerHide

   Answer$\displaystyle 2^{3} = 8$

   Steps

   1. $\displaystyle \log_{2}8$$\displaystyle = 3$
   2. In exponential form
   3. $\displaystyle 2^{3}$$\displaystyle = 8$
3. 03What number goes in □?$\displaystyle \log_{a}1 = \square$AnswerHide

   Answer$\displaystyle 0$

   Steps

   1. $\displaystyle a^{0}$$\displaystyle = 1$
   2. In logarithmic form
   3. $\displaystyle \log_{a}1$$\displaystyle = 0$
4. 04What is the value?$\displaystyle \log_{3}27$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. $\displaystyle 3^{3}$$\displaystyle = 27$
   2. In logarithmic form
   3. $\displaystyle \log_{3}27$$\displaystyle = 3$
5. 05Write in logarithmic form:$\displaystyle 5^{3} = 125$AnswerHide

   Answer$\displaystyle \log_{5}125 = 3$

   Steps

   1. $\displaystyle 5^{3}$$\displaystyle = 125$
   2. In logarithmic form
   3. $\displaystyle \log_{5}125$$\displaystyle = 3$
6. 06Given $\log_{2}4 = 2$, evaluate:$\displaystyle \log_{2}2 + \log_{2}4$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. $\displaystyle 2^{1}$$\displaystyle = 2$
   2. Adding exponents means multiplying
   3. $\displaystyle 2 \times 4$$\displaystyle = 2^{3}$
   4. In logarithmic form
   5. $\displaystyle \log_{2}(2 \times 4)$$\displaystyle = 3$

FAQ