Logarithms – log laws and exponential equations

What does a logarithm mean?

Since 10³ = 1,000, the base-ten logarithm of 1,000 is 3. We write lg(1,000) = 3. The logarithm tells us which exponent the base ten should have.

For real logarithms, a must be positive. The base b must be positive and cannot be one. The common logarithm has base ten. The natural logarithm, ln, has base e.

There is no real logarithm of zero or a negative number. However, the value of the logarithm itself can be negative: lg(0.01) = −2 because 10⁻² = 0.01.

Revise powers and index laws if the relationship between base and exponent is unclear.

Three log laws with examples

For the same valid base, positive a and b and a real number r, the following apply:

• Product: log(ab) = log(a) + log(b).
• Quotient: log(a/b) = log(a) − log(b).
• Power: log(a^r) = r · log(a).

Base-ten example: lg(100 · 10) = lg(100) + lg(10) = 2 + 1 = 3. For the quotient, lg(100/10) = 2 − 1 = 1.

The power rule lets you move an exponent in front of the logarithm. This is useful when the exponent is unknown.

Solve an exponential equation

In 2^x = 10, we want to find x in the exponent. Take the base-ten logarithm of both sides and use the power rule.

1. lg(2^x) = lg(10).
2. x · lg(2) = 1.
3. x = 1/lg(2) ≈ 3.322.

The value lies between three and four. This is reasonable because 2³ = 8 and 2⁴ = 16. Use the unrounded value in your calculator when checking the answer.

For a^x = b, the same method can be written as x = lg(b)/lg(a), when a > 0, a ≠ 1 and b > 0. You can also use ln in both the numerator and denominator.

Practise logarithms with answers

Problem 1

Calculate lg(100) and lg(0.001).

Problem 2

Solve 10^x = 500. Round to three decimal places.

Problem 3

Solve 3 · 2^x = 24.