The logarithm

Exponential and logarithmic functions, grade 12 (5 units) Lesson 3 of 23Learning

The logarithm

An exponent that is no whole number

The equation $\displaystyle {2^x}={8}$ has the solution $\displaystyle {x}={3}$ , because $\displaystyle {2^3}={8}$ . The equation $\displaystyle {2^x}={7}$ has one solution too, because $\displaystyle 2^x$ increases and takes every positive value:

$\displaystyle {2^2}={4}$ and $\displaystyle {2^3}={8}$ , so the solution is between 2 and 3.
It cannot be written as a fraction or a root, so it is given a name: $\displaystyle \log_2 7$ – the logarithm of 7 to the base 2.

The definition of the logarithm

The logarithm of $\displaystyle b$ to the base $\displaystyle a$ is the exponent to which $\displaystyle a$ is raised to give $\displaystyle b$ :

\displaystyle \log_{a}b=x\iff a^{x}=b

\displaystyle a>0,\ a\neq1,\ b>0

The base is positive and not 1, as in the exponential function.
$\displaystyle b$ is positive: $\displaystyle a^x$ is positive for every $\displaystyle x$ , so $\displaystyle \log_2 0$ and $\displaystyle \log_2(-4)$ have no value.

Calculating a logarithm

To calculate a logarithm, write the number as a power of the base:

$\displaystyle {\log_{2}32}={5}$ , because $\displaystyle {2^{5}}={32}$ .
$\displaystyle {\log_{\frac{1}{3}}9}={-2}$ , because $\displaystyle {\left(\frac{1}{3}\right)^{-2}}={9}$ .
$\displaystyle {\log_{5}\sqrt{5}}={\frac{1}{2}}$ , because $\displaystyle {5^{\frac{1}{2}}}={\sqrt{5}}$ .
$\displaystyle {\log_{4}8}={\frac{3}{2}}$ , because $\displaystyle {4^{\frac{3}{2}}}={\left(\sqrt{4}\right)^{3}}={8}$ .

Properties that follow from the definition

The logarithm and the power to the same base undo each other:

\displaystyle \log_{a}(a^{b})

\displaystyle =

\displaystyle b

\displaystyle a^{\log_{a}x}

\displaystyle =

\displaystyle x

\displaystyle x>0

$\displaystyle {\log_a 1}={0}$ , because $\displaystyle {a^0}={1}$ .
$\displaystyle {\log_a a}={1}$ , because $\displaystyle {a^1}={a}$ .

The logarithm to base 10

The logarithm to base 10 is written without the base: $\displaystyle {\log x}={\log_{10}x}$ . A calculator has a $\displaystyle \log$ key for it.

$\displaystyle {\log1000}={3}$ and $\displaystyle {\log0.01}={-2}$ .
$\displaystyle \log50\approx1.699$ : $\displaystyle 10^{1.699}$ is close to 50.

Solving

\displaystyle {a^x}={b}

By the definition, the solution of $\displaystyle {a^x}={b}$ is $\displaystyle {x}={\log_a b}$ . The example below solves $\displaystyle {4\cdot3^{x}}={20}$ :

Isolate the power: $\displaystyle {3^{x}}={5}$ .
By the definition: $\displaystyle {x}={\log_{3}5}$ .
On a calculator with a $\displaystyle \log_{\square}$ key: $\displaystyle x\approx1.465$ . How to calculate it with the $\displaystyle \log$ key alone – in the next lesson.

Exercise 1

Calculate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Calculate with the properties that follow from the definition.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Solve by the definition of the logarithm.

Exercise 4

Solve.
Write the solution with a logarithm if it is no whole number or fraction.

Formulas: exponential and logarithmic functions, grade 12 (5 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
Change of base
$\displaystyle \log_{c}x$ $\displaystyle =$ $\displaystyle \frac{\log_{a}x}{\log_{a}c}$
$\displaystyle \log_{a}b$ $\displaystyle =$ $\displaystyle \frac{1}{\log_{b}a}$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

The exponential and the logarithmic function

Every exponential as a power of $\displaystyle e$
$\displaystyle a^{x}$ $\displaystyle =$ $\displaystyle e^{x\ln a}$
$\displaystyle a^x$ and $\displaystyle \log_a x$ – inverse functions
$\displaystyle y=a^{x}\iff x=\log_{a}y$ $\displaystyle y>0$
The number $\displaystyle e$
$\displaystyle e\approx2.718$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$
The doubling time $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle 2$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (a^{x})^{\prime}$ $\displaystyle =$ $\displaystyle a^{x}\cdot\ln a$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
$\displaystyle (\log_{a}x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x\cdot\ln a}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals, the area and accumulation

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int a^{x}\,dx=\frac{a^{x}}{\ln a}+C$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
Where $\displaystyle f(x)\neq0$
$\displaystyle \int\frac{f^{\prime}(x)}{f(x)}\,dx=\ln|f(x)|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The accumulation function’s derivative
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$