Converting from Logarithmic to Exponential Form

A logarithm is an exponent. That is, …

loga y = exponent to which the base a must be raised to obtain y

In other words, loga y = x is equivalent to ax = y

Example 1 Write the logarithmic equation log3 (9) = 2 in

equivalent exponential form.

( ) =

Converting from Logarithmic to Exponential Form

Example 2 Write the logarithmic equation C = log

H (A) in

equivalent exponential form.

= ( )

Converting from Logarithmic to Exponential Form

loga y = x is equivalent to ax = y

Example 3 Write the logarithmic equation log (10,000) = 4 in

equivalent exponential form.

= ( )

Converting from Logarithmic to Exponential Form

Common Logarithm: log10

y = log y

Example 4 Write the logarithmic equation A = ln (19) in

equivalent exponential form.

( ) =

Converting from Logarithmic to Exponential Form

Natural Logarithm: loge y = ln y

(Note: e у�2.718)

Converting from Exponential to Logarithmic Form

Example 1 Write the exponential equation 8 = 23 in equivalent logarithmic form.

log ( ) =

Converting from Exponential to Logarithmic Form

loga y = x is equivalent to ax = y

Example 2 Write the exponential equation MK = D in equivalent logarithmic form.

log ( ) =

Converting from Exponential to Logarithmic Form

loga y = x is equivalent to ax = y

Evaluating Logarithms

Example 1 Determine the value of the logarithmic expression.

(a) log2 16 = (b) log 100 = (c) ln e =

Evaluating Logarithms loga y = exponent to which the base a must be raised to obtain y Note: loga y = x is equivalent to ax = y

Example 2 Determine the value of the logarithmic expression.

(a) (b) (c)

Evaluating Logarithms Properties of Exponents:

z0a = 1, a 0 -mm1a =

a§ ·¨ ¸© ¹

-mm1 = a

a

§ ·¨ ¸© ¹

31log =

81

ln 1 =

1/2log 2 =

Example 3 Determine the value of the logarithmic expression.

(a) (b) (c)

Evaluating Logarithms Properties of Exponents:

z0a = 1, a 0 -mm1a =

a§ ·¨ ¸© ¹

-mm1 = a

a1/2a = a

log 0.01 =

6log 6 =

16log 4 =

Simplifying Logarithmic and Exponential Expressions

Example 1 Simplify the following expression.

Simplifying Logarithmic and Exponential Expressions

loga y = exponent to which the base a must be raised to obtain y Note: loga y = x is equivalent to ax = y

43 3log (3 ) + log (3) =

Example 2 Simplify the following expressions.

(a) (b)

Simplifying Logarithmic and Exponential Expressions

loga y = exponent to which the base a must be raised to obtain y Note: loga y = x is equivalent to ax = y

2log 82 =

5ln (e )e =