1. Common logarithms

2. What you’ll learn
Solve exponential equations and inequalities using common logarithms
Evaluate logarithmic expressions using the Change of Base Formula

3. 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.

4. Common logarithms
Common logarithms are usually written without the subscript 10
Ex) log10 x = log x, x > 0

5. 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 a
Ex) log5 = log10 12 / log 10 5

6. 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 ≈

7. 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

8. Solving Exponential Equations
Solve 3x = 11
3x = 11
Log 3x = log 11
x log 3 = log 11
x = log 11 / log 3
x ≈ /
x ≈