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Logarithms – Solving exponential equations

📘 Notes

Logarithms — Solving Exponential Equations

Exponential equations are equations in which the unknown appears in the power. They can often be solved by writing both sides with the same base or by using logarithms.

1. Main idea

In an exponential equation, the variable is in the exponent.

\[ 2^x=8,\qquad 3^{2x-1}=7,\qquad 5^{x+1}=2^{x} \]

There are two main methods:

  • rewrite both sides using the same base
  • take logarithms of both sides

2. Method 1: Write both sides with the same base

If both sides can be written using the same base, then equate the powers.

\[ a^m=a^n \quad \Longrightarrow \quad m=n \]

Example: Solve

\[ 2^{x+1}=16 \]

Write \(16\) as a power of 2:

\[ 2^{x+1}=2^4 \]

Therefore:

\[ x+1=4 \quad \Rightarrow \quad x=3 \]

3. Another same-base example

Example: Solve

\[ 9^{x}=3^{2x+1} \]

Since \(9=3^2\),

\[ (3^2)^x=3^{2x+1} \] \[ 3^{2x}=3^{2x+1} \]

This would give \[ 2x=2x+1, \] which is impossible.

\[ \text{No solution} \]

4. Method 2: Use logarithms

If the equation cannot easily be written using the same base, take logarithms of both sides.

\[ a^x=b \quad \Longrightarrow \quad \log(a^x)=\log b \]

Then use the law:

\[ \log(a^x)=x\log a \]

Example: Solve

\[ 3^x=7 \]

Take logarithms:

\[ \log(3^x)=\log 7 \] \[ x\log 3=\log 7 \]

Therefore:

\[ x=\frac{\log 7}{\log 3} \]

5. Worked example with algebra

Example: Solve

\[ 2^{x+2}=5 \]

Take logarithms:

\[ \log(2^{x+2})=\log 5 \] \[ (x+2)\log 2=\log 5 \]

So:

\[ x+2=\frac{\log 5}{\log 2} \] \[ x=\frac{\log 5}{\log 2}-2 \]

6. Example with different bases on both sides

Example: Solve

\[ 5^{x}=2^{x+1} \]

Take logarithms:

\[ \log(5^x)=\log(2^{x+1}) \] \[ x\log 5=(x+1)\log 2 \]

Expand:

\[ x\log 5=x\log 2+\log 2 \] \[ x(\log 5-\log 2)=\log 2 \]

Therefore:

\[ x=\frac{\log 2}{\log 5-\log 2} \]

7. Summary of methods

Type of equation Best method
\(2^x=32\) Rewrite with the same base
\(3^x=7\) Use logarithms
\(5^x=2^{x+1}\) Use logarithms

8. Exam tips

  • First check whether both sides can be written in the same base.
  • If not, take logarithms of both sides.
  • Use \(\log(a^x)=x\log a\).
  • Keep algebra neat when collecting terms in \(x\).
  • Exact answers in logarithmic form are often accepted unless decimals are requested.