The big idea: If both sides can be written with the same base, then the exponents must be equal.
Worked example
Solve 2x+1 = 16.
Step by step
- Rewrite 16 as a power of 2.
- Set exponents equal.
- Solve.
Final answer
x = 3
Use common bases when you can: This is often the fastest method and avoids unnecessary calculator work.
When bases do not match: If you cannot rewrite both sides with a common base, use logarithms to bring the exponent down.
Worked example
Solve 3x = 20.
Step by step
- Take logs of both sides.
- Use the power law.
- Solve for x.
Final answer
x ≈ 2.727
Any consistent log base works: You can use log or ln, as long as you use the same one on both sides.
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The big idea: A logarithmic equation can often be rewritten as an exponential equation.
Worked example
Solve log2 x = 5.
Step by step
- Rewrite in exponential form.
- Calculate.
Final answer
x = 32
Second example
Solve log10 x = 2.3.
Step by step
- Rewrite in exponential form.
- Use a calculator.
Final answer
x ≈ 199.53
Logs need positive inputs: You cannot take the logarithm of 0 or a negative number in the real-number setting used here.
Worked example
Why can log x = 2 never have x = -100 as a solution?
Step by step
- Because the input of a logarithm must be positive.
- -100 is negative, so log(-100) is not valid in this context.
Final answer
x must be positive.
Always check sense: After solving, ask whether the answer is valid in the original equation. This matters especially with logarithms.