10-4 Common logarithms

What you’ll learn

Solve exponential equations and inequalities using common logarithms

Evaluate logarithmic expressions using the Change of Base Formula

Key words

Common logarithms : logarithms that use 10 as the base

Change of Base Formula : allows you to write equivalent logarithmic expressions that have different bases.

Common logarithms

Common logarithms are usually written without the subscript 10

Ex) log10 x = log x, x > 0

Change of Base Formula

For all positive numbers, a, b and n, where a and b doesn’t equal to 1,

log a n = log b n / log b aEx) log5 = log10 12 / log 10 5

Change of Base Formula example

Express log4 25 in terms of common logarithms. Then approximate its value to four decimal places.

Log4 25= log10 25 / log10 4

≈ 2.3219

Proving the formula

To prove the change of base formula, let log a n = x,

ax = n Log b ax = log b n

x log b a = log b n

x = log b n / log b a

Log a n = log b n / log b a

Solving Exponential Equations

Solve 3x = 11 3x = 11 Log 3x = log 11 x log 3 = log 11 x = log 11 / log 3 x ≈ 1.0414 / 0.4771 x ≈ 2.1828