Logarithms and Exponential Equations: Complete Guide

If you are looking for Logarithms and Exponential functions, This guide covers exponential functions, logarithmic functions, solving equations of both types, and real-world applications. Work through every example here and you will be ready for both classwork and national exams.

Exponential and Logarithmic Functions

What Is an Exponential Function?

An exponential function has the form $f (x) = a^{x}$ where $a > 0$ and $a \neq 1$ . The base $a$ is a constant and $x$ is the variable in the exponent — that is what makes it "exponential."

Two bases come up constantly in S5 and beyond. Base 10 gives the common logarithm. Euler's number $e \approx 2.71828$ gives the natural exponential function $f (x) = e^{x}$ , which appears in science, economics, and population modelling.

Key Properties of Exponential Functions

For $f (x) = a^{x}$ with $a > 0, a \neq 1$ :

Domain is all real numbers: $x \in R$
Range is strictly positive: $f (x) > 0$ for all $x$
The graph always passes through $(0, 1)$ because $a^{0} = 1$
When $a > 1$ : the function increases (exponential growth)
When $0 < a < 1$ : the function decreases (exponential decay)
The x-axis is a horizontal asymptote — the graph approaches it but never touches it

What Is a Logarithmic Function?

The logarithm is the inverse of the exponential. If $a^{y} = x$ then we write $y = \log_{a} x$ . In plain language, the logarithm answers: what power do I raise $a$ to in order to get $x$ ?

Example: $\log_{2} 8 = 3$ because $2^{3} = 8$ .

The two most common forms:

\log_{10} x = \log x (common log — base written without subscript)

\log_{e} x = \ln x (natural log)

The Four Log Rules You Must Know

Product Rule:

\log_{a} (m n) = \log_{a} m + \log_{a} n

Quotient Rule:

\log_{a} (\frac{m}{n}) = \log_{a} m - \log_{a} n

Power Rule:

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

Change of Base:

\log_{a} b = \frac{\ln b}{\ln a} = \frac{\log b}{\log a}

The change of base formula is what you use when your calculator only has $\log$ and $\ln$ but the question uses a different base.

Worked Example 1: Evaluating a Logarithm

Evaluate $\log_{3} 81$ .

Ask: what power of 3 gives 81?

3^{4} = 81 ⟹ \log_{3} 81 = 4

Worked Example 2: Expanding Using Log Rules

Expand $\log_{2} (\frac{8 x^{3}}{y})$ .

\log_{2} (\frac{8 x^{3}}{y}) = \log_{2} 8 + \log_{2} x^{3} - \log_{2} y = 3 + 3 \log_{2} x - \log_{2} y

Worked Example 3 — Change of Base

Evaluate $\log_{5} 200$ using a calculator.

\log_{5} 200 = \frac{\log 200}{\log 5} = \frac{2.3010}{0.6990} \approx 3.292

Properties of Logarithmic Graphs

Since $\log_{a} x$ is the inverse of $a^{x}$ , their graphs are reflections across the line $y = x$ . For $f (x) = \log_{a} x$ :

Domain: $x > 0$ — you cannot log zero or a negative number
Range: all real numbers
Passes through $(1, 0)$ because $\log_{a} 1 = 0$ for any base
The y-axis is a vertical asymptote

3.2 Solving Exponential and Logarithmic Equations

Method 1 : Same Base (Exponential Equations)

When both sides share the same base, set the exponents equal.

Example 4: Solve $2^{x + 1} = 32$ .

2^{x + 1} = 2^{5} ⟹ x + 1 = 5 ⟹ x = 4

Example 5: Solve $9^{x} = 27$ .

Write both sides as powers of 3:

3^{2 x} = 3^{3} ⟹ 2 x = 3 ⟹ x = \frac{3}{2}

Method 2 : Taking Logarithms (Exponential Equations)

When you cannot match bases, apply $\log$ or $\ln$ to both sides.

Example 6: Solve $3^{x} = 20$ .

x \log 3 = \log 20 ⟹ x = \frac{\log 20}{\log 3} = \frac{1.3010}{0.4771} \approx 2.727

Example 7: Solve $5^{2 x - 1} = 100$ .

(2 x - 1) \log 5 = \log 100 = 2

2 x - 1 = \frac{2}{\log 5} = \frac{2}{0.6990} \approx 2.861

x \approx \frac{3.861}{2} \approx 1.930

Method 3: Convert to Exponential Form (Log Equations)

If $\log_{a} x = k$ , then $x = a^{k}$ .

Example 8: Solve $\log_{2} x = 5$ .

x = 2^{5} = 32

Example 9: Solve $\log_{3} (x - 2) = 4$ .

x - 2 = 3^{4} = 81 ⟹ x = 83

Check: $x - 2 = 81 > 0$ ✓

Example 10 — Combining Log Terms First

Solve $\log x + \log (x - 3) = 1$ .

\log [x (x - 3)] = 1 ⟹ x (x - 3) = 10

x^{2} - 3 x - 10 = 0 ⟹ (x - 5) (x + 2) = 0

Reject $x = - 2$ since $\log (- 2)$ is undefined. Answer: $x = 5$ .

Example 11 — Natural Log Equation

Solve $\ln (x + 1) - \ln x = \ln 3$ .

\ln (\frac{x + 1}{x}) = \ln 3 ⟹ \frac{x + 1}{x} = 3 ⟹ x + 1 = 3 x ⟹ x = \frac{1}{2}

Example 12 — Quadratic in Disguise

Solve $e^{2 x} - 3 e^{x} - 10 = 0$ .

Let $u = e^{x}$ :

u^{2} - 3 u - 10 = 0 ⟹ (u - 5) (u + 2) = 0

Since $e^{x} > 0$ always, reject $u = - 2$ . Then:

e^{x} = 5 ⟹ x = \ln 5 \approx 1.609

3.3 Real-World Applications

Compound Interest

Money compounding at rate $r$ over $t$ years:

A = P {(1 + \frac{r}{n})}^{n t}

For continuous compounding:

A = P e^{r t}

Example 13: 500,000 RWF invested at 8% per year compounded monthly for 5 years.

A = 500,000 {(1 + \frac{0.08}{12})}^{60} \approx 500,000 \times 1.4898 \approx 744,912 RWF

Example 14: How long to double at 6% continuous interest?

2 = e^{0.06 t} ⟹ \ln 2 = 0.06 t ⟹ t = \frac{\ln 2}{0.06} \approx 11.55 years

This is the basis of the Rule of 70 used in economics: divide 70 by the interest rate to estimate doubling time.

Population Growth

Any quantity growing at a rate proportional to its size follows:

N (t) = N_{0} e^{k t}

where $N_{0}$ is the initial amount and $k > 0$ for growth, $k < 0$ for decay.

Example 15: A bacterial culture starts with 500 cells and doubles every 3 hours.

(a) Find $k$ : when $t = 3$ , $N = 1000$ :

2 = e^{3 k} ⟹ k = \frac{\ln 2}{3} \approx 0.2310 per hour

(b) Population after 9 hours:

N = 500 e^{0.2310 \times 9} = 500 \times 8 = 4000 cells

\ln 100 = 0.2310 t ⟹ t = \frac{\ln 100}{0.2310} \approx 19.94 hours

Radioactive Decay and Half-Life

Radioactive material decays as $N (t) = N_{0} e^{- k t}$ . The half-life is:

t_{1 / 2} = \frac{\ln 2}{k}

Example 16: Carbon-14 has a half-life of 5,730 years. A sample contains 30% of its original carbon-14. How old is it?

Find $k$ :

k = \frac{\ln 2}{5730} \approx 0.0001209 per year

Solve for $t$ :

0.30 = e^{- 0.0001209 t}

\ln (0.30) = - 0.0001209 t

t = \frac{- \ln (0.30)}{0.0001209} = \frac{1.2040}{0.0001209} \approx 9,958 years

This is exactly how archaeologists use carbon dating to determine the age of ancient objects.

The Richter Scale

Earthquake magnitude uses a base-10 logarithm:

M = \log (\frac{I}{I_{0}})

Each step up the scale means 10 times the intensity. A magnitude 8.2 earthquake compared to magnitude 6.0:

\frac{I_{B}}{I_{A}} = 10^{8.2 - 6.0} = 10^{2.2} \approx 158 times more intense

Sound — The Decibel Scale

β = 10 \log (\frac{I}{I_{0}}) dB, I_{0} = 10^{- 12} {W/m}^{2}

Example 18: A conversation at 60 dB:

6 = \log (\frac{I}{10^{- 12}}) ⟹ I = 10^{- 6} {W/m}^{2}

That is one million times the threshold of human hearing.

pH in Chemistry

pH = - \log [H^{+}]

Example 19: Find pH when $[H^{+}] = 3.2 \times 10^{- 4}$ mol/L.

pH = - \log (3.2 \times 10^{- 4}) = - (0.5051 - 4) = 3.49 (acidic)

Summary Table

Concept	Formula
Definition	$a^{y} = x ⟺ y = \log_{a} x$
Product rule	$\log_{a} (m n) = \log_{a} m + \log_{a} n$
Quotient rule	$\log_{a} (m / n) = \log_{a} m - \log_{a} n$
Power rule	$\log_{a} (m^{n}) = n \log_{a} m$
Change of base	$\log_{a} b = \ln b / \ln a$
Compound interest	$A = P (1 + r / n)^{n t}$
Continuous growth/decay	$N (t) = N_{0} e^{k t}$
Half-life	$t_{1 / 2} = \ln 2 / k$

Practice Questions

Evaluate $\log_{4} 64$
Simplify $\log_{5} 125 + \log_{5} \frac{1}{5}$
Solve $4^{x - 1} = 8^{x}$
Solve $\log_{2} (x + 4) + \log_{2} (x - 4) = 4$
A population of 2,000 grows continuously at 5% per year. When does it reach 10,000?
Blood pH is 7.4. Find the hydrogen ion concentration $[H^{+}]$ .

If any step here needs more explanation, our Mathematics tutors on Mathrone can work through problems with you in a live session or send us a message on +250786684285. You can also visit our Mathematics course for structured practice that follows the REB syllabus .

The topic that connects directly to this unit is differential calculus specifically the derivatives of $e^{x}$ and $\ln x$ . Every rule you practised here feeds into that next step.

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