The laws of logarithms and the change of base

Exponential and logarithmic functions, grade 12 (5 units) Lesson 4 of 23Learning

The laws of logarithms and the change of base

The logarithm of a product

A logarithm is an exponent, so the laws of powers give laws of logarithms. When powers are multiplied their exponents are added – so the logarithms are added:

\displaystyle \log_{a}x+\log_{a}y

\displaystyle =

\displaystyle \log_{a}(x\cdot y)

\displaystyle x>0,\ y>0

Let $\displaystyle {m}={\log_a x}$ and $\displaystyle {n}={\log_a y}$ , that is $\displaystyle {x}={a^m}$ and $\displaystyle {y}={a^n}$ .
$\displaystyle {x\cdot y}={a^m\cdot a^n}={a^{m+n}}$ , so $\displaystyle {\log_a(x\cdot y)}={m+n}$ .

A quotient, a power and a root

In the same way, the quotient law and the law of a power of a power give:

\displaystyle \log_{a}x-\log_{a}y

\displaystyle =

\displaystyle \log_{a}\left(\frac{x}{y}\right)

\displaystyle x>0,\ y>0

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

\displaystyle =

\displaystyle b\cdot\log_{a}x

\displaystyle x>0

A root is a power, so $\displaystyle {\log_a\sqrt[n]{x}}={\frac{1}{n}\log_a x}$ .
$\displaystyle {\log_a\frac{1}{x}}={-\log_a x}$ .
There is no law for the logarithm of a sum: $\displaystyle \log_a(x+y)$ is not $\displaystyle \log_a x+\log_a y$ .

Worked example: a calculation

The example below calculates $\displaystyle \log_{2}40-2\log_{2}\sqrt{5}$ by joining the logarithms into one:

$\displaystyle {2\log_{2}\sqrt{5}}={\log_{2}(\sqrt{5})^{2}}={\log_{2}5}$ .
$\displaystyle {\log_{2}40-\log_{2}5}={\log_{2}\frac{40}{5}}={\log_{2}8}={3}$ .

The change of base

A logarithm to one base is a quotient of two logarithms to any other base $\displaystyle c$ :

\displaystyle \log_{a}x

\displaystyle =

\displaystyle \frac{\log_{c}x}{\log_{c}a}

\displaystyle c>0,\ c\neq1

Let $\displaystyle {y}={\log_a x}$ , that is $\displaystyle {a^y}={x}$ .
Take the logarithm to the base $\displaystyle c$ of both sides: $\displaystyle {y\cdot\log_c a}={\log_c x}$ , so $\displaystyle {y}={\frac{\log_c x}{\log_c a}}$ .
$\displaystyle {\log_{8}32}={\frac{\log_{2}32}{\log_{2}8}}={\frac{5}{3}}$ .
With $\displaystyle {c}={b}$ you get $\displaystyle {\log_a b}={\frac{1}{\log_b a}}$ : for example $\displaystyle {\log_{9}3}={\frac{1}{2}}$ and $\displaystyle {\log_{3}9}={2}$ .

A logarithm to any base on a calculator

Changing to base 10 calculates any logarithm with the $\displaystyle \log$ key:

\log_{3}20=\frac{\log20}{\log3}\approx2.727

Round only at the end, after dividing.
A check: $\displaystyle 3^{2.727}$ is close to 20.

Expressing with a parameter

It is given that $\displaystyle {\log_{2}5}={a}$ .
The example below expresses other logarithms with $\displaystyle a$ , through the numbers 2 and 5 in them:

$\displaystyle {\log_{2}40}={\log_{2}(8\cdot5)}={\log_{2}8+\log_{2}5}={3+a}$ .
$\displaystyle {\log_{2}0.2}={\log_{2}\frac{1}{5}}={-a}$ .
$\displaystyle {\log_{5}2}={\frac{1}{\log_{2}5}}={\frac{1}{a}}$ .

Exercise 1

Calculate with the laws of logarithms.

/ – 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 change of base.

/ – 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

Write as a quotient of logarithms to base 10, and calculate on a calculator.
Round to three decimal places, in a line with $\displaystyle \approx$ .

/ – 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 4

It is given that $\displaystyle {\log_{2}3}={a}$ .
In parts 1–3 write each logarithm with $\displaystyle \log_{2}3$ and numbers.

/ – 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

Write the three results with $\displaystyle a$ .
Express $\displaystyle \log_{3}2$ with $\displaystyle a$ too.

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
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$