Change of Base Formula

Solve for x:  2^x = 10

This cannot be solved in the manner other exponentials have been solved in previous lessons.

• Start by taking the log or ln of both sides.  Log or ln is chosen because they are the two keys on your calculator.
• 2^x = 10                                   
• ln 2^x = ln 10
• x ln 2 = ln 10
• x = ln 10
        ln 2
• Another way to solve this equation is simply change the exponential function to a logarithmic function.
• 2^x = 10
• x = ln₂10
• Since both solutions yield a value for x,
• ln₂10 = ln 10        This is called the "Change of Base Formula"
                ln 2

The change of base formula can be useful when graphing y = ln₃x in a graphing calculator.  Simply change the expression to y = ln x    or    y = log x   and enter it into Y1.
                                            ln 3                 log 3

Problems