# Exponentials and Logarithms

## Exponentials and Logarithms Revision

**Exponentials and Logarithms**

A sum such as 4+3=7 has **two inverses**: 7-3=4 and 7-4=3.

A product such as 8\times6=48 has **two inverses**: 48\div6=8 and 48\div8=6.

An **exponential**** (power)** such as 3^{4}=81 has an **inverse** of the fourth root: \sqrt[4]{81}=3. But for the pattern to continue there **must be another inverse** – an operation involving 81 and 3 to get back to 4.

**This other inverse is the logarithm.**

**Logarithms are the Inverse of Exponentials**

\log_{a}(b)=c means that a^{c}=b

a^{1}=a for any a so \log_{a}(a)=1

a^{0}=1 for any a so \log_{a}(1)=0

**Example: **\log_{3}(81)=4 because 3^{4}=81

We also have other rules:

\log_{a}(x)+\log_{a}(y)=\log_{a}(xy)

\log_{a}(x)-\log_{a}(y)=\log_{a}\left(\dfrac{x}{y}\right)

\log_{a}(x^{n})=n\log_{a}(x)

\log_{a}(x)=\dfrac{\log_{b}(x)}{\log_{b}(a)}

**Note:** The little number after the \text{log} is called the **base**. The most common base is 10, but this is usually left out – i.e. for \text{log}_{10} we just write \text{log} .

**Using Logarithms and Exponentials to Solve Equations**

We can use the rules above – called the** laws of logarithms** – to solve **equations that involve exponentials**.

**Example: **3^{4x}=5

\log(3^{4x})=\log(5)

4x\log(3)=\log(5)

\begin{aligned}x&=\dfrac{\log(5)}{4\log(3)}\\[1.2em]&=0.366\end{aligned}

**Note: **The **base of the logarithm** was **not given** in this example. This is because we would get the **same answer using any base**.

On the flipside, we can use **exponentials** to **solve equations involving logarithms**.

**Example: **6\log_{4}(x)=17

\log_{4}(x)=\dfrac{17}{6}

\begin{aligned}x&=4^{\frac{17}{6}}\\[1.2em]&=50.8\end{aligned}

**Graphs of Exponentials**

An **exponential** **graph** is a graph of the function y=a^{x} for some a>0. They all have **the same basic shape**. If a>1 then y increases as x increases. If a<1 then y decreases as x increases. Larger a gives faster increase if a>1 while smaller a gives faster decrease if a<1.

**Note: Exponential graphs never reach** \mathbf{0}.

On the left graph is y=2^{x}, y=3^{x} and y=4^{x}.

On the right graph is y=\left(\dfrac{1}{2}\right)^{x}, y=\left(\dfrac{1}{3}\right)^{x} and y=\left(\dfrac{1}{4}\right)^{x}.

## Exponentials and Logarithms Example Questions

**Question 1: **Write the following in logarithm notation:

a) 7^{3}=343

b) 5^{4}=625

c) 2^{16}=65536

d) 729^{\frac{1}{3}}=9

e) 22^{0}=1

**[5 marks]**

a) \log_{7}(343)=3

b) \log_{5}(625)=4

c) \log_{2}(65536)=16

d) \log_{729}(9)=\dfrac{1}{3}

e) \log_{22}(1)=0

**Question 2: **Evaluate:

a) \log_{2}(16)

b) \log_{5}(125)

c) \log_{3}(243)

d) \log_{\frac{1}{4}}\left(\dfrac{1}{16}\right)

**[4 marks]**

a) 4 because 2^{4}=16

b) 3 because 5^{3}=125

c) 5 because 3^{5}=243

d) 2 because \left(\dfrac{1}{4}\right)^{2}=\dfrac{1}{16}

**Question 3: **Write these expressions as a single logarithm:

a) \log_{a}(12)+\log_{a}(6)

b) 2\log_{b}(5)+\log_{b}(4)

c) 3\log_{6}(2)+\dfrac{1}{2}\log_{6}(9)

**[3 marks]**

a)

\begin{aligned}\log_{a}(12)+\log_{a}(6)&=\log_{a}(12\times6)\\[1.2em]&=\log_{a}(72)\end{aligned}

b)

\begin{aligned}2\log_{b}(5)+\log_{b}(4)&=\log_{b}(5^{2})+\log_{b}(4)\\[1.2em]&=\log_{b}(25)+\log_{b}(4)\\[1.2em]&=\log_{b}(25\times4)\\[1.2em]&=\log_{b}(100)\end{aligned}

c)

\begin{aligned}3\log_{6}(2)+\dfrac{1}{2}\log_{6}(9)&=\log_{6}(2^{3})+\log_{6}(9^{\frac{1}{2}})\\[1.2em]&=\log_{6}(8)+\log_{6}(3)\\[1.2em]&=\log_{6}(8\times3)\\[1.2em]&=\log_{6}(24)\end{aligned}**Question 4: **Solve 6^{7x}=38, giving your answer to 3 significant figures.

**[2 marks]**

6^{7x}=38

7x=\log_{6}(38)

x=\dfrac{1}{7}\log_{6}(38)

x=0.290

**Question 5:** Solve \log_{6}(x-2)+\log_{6}(x-7)=2

**[4 marks]**

\log_{6}(x-2)+\log_{6}(x-7)=2

\log_{6}((x-2)(x-7))=2

(x-2)(x-7)=6^{2}

x^{2}-9x+14=36

x^{2}-9x-22=0

(x-11)(x+2)=0

x=11 or x=-2