# Logarithmic Equations: Introduction and Simple Equations

A logarithmic function is the inverse to an exponential function. Just like exponential function have common bases and a natural base; logarithmic functions have common logs and a natural log.

This discussion will focus on the common logarithmic functions.

The general common logarithmic equation is:

COMMON LOGARITHMIC FUNCTION

if and only if     x = ay

Where a > 0, a ≠ 1 and x > 0

When reading say, "log base a of x".

Some examples are:

1. because 102 = 100

2. because 34 = 81

3. because 152 = 225

Notice in the examples that the base of the log is also the base of the corresponding exponent. In example 1 above, the logarithmic function has a log of base 10 and the corresponding exponential function has a base of 10.

If you see log with no base is means log of base 10 or log = log10.

Some basic properties of logarithmic functions are:

Property 1: because a0 = 1

Property 2: because a1 = a

Property 3: If $lo{g}_{a}x=lo{g}_{a}y$, then x = y       One-to One Property

Property 4: and ${a}^{{\text{log}}_{a}x}=x$           Inverse Property

Let's solve some simple logarithmic equations:

log x = 4
 Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the log equals neither 0 nor 1. Property 3 does not apply since a log is not set equal to a log of the same base. Therefore Property 4 is the most appropriate. Property 4 - Inverse Step 2: Apply the Property. Remember $log=lo{g}_{10}$. Since the log has a base of 10, taking the inverse means to rewrite both sides as exponents with base 10. log x = 4 Original 10logx = 104 Exponent of 10 Step 3: Solve for x. Property 4 states that ${a}^{lo{g}_{a}x}=x$, therefore the left-hand side becomes x. x = 104   Apply Property x = 10,000 Evaluate
Example 1:
 Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the log equals neither 0 nor 1. Since a log is set equal to a log of the same base. Property 3 is the most appropriate. Property 3 - One to One Step 2: Apply the Property. Property 3 states that if $lo{g}_{a}x=lo{g}_{a}y$, then x = y. Therefore x = 4x - 9. x = 4x - 9 Apply Property Step 3: Solve for x. -3x = -9    Subtract 4x x = 3        Divide by -3
Example 2:
 Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the log equals neither 0 nor 1. Property 3 does not apply since a log is not set equal to a log of the same base. Therefore Property 4 is the most appropriate. Property 4 - Inverse Step 2: Apply the Property. Since the log has a base of 3, taking the inverse means to rewrite both sides as exponents with base 3. Original ${3}^{{\mathrm{log}}_{3}3x}={3}^{5}$ Exponent of 3 Step 3: Solve for x. Property 4 states that ${a}^{lo{g}_{a}x}=x$, therefore the left-hand side becomes x. 3x = 35       Apply Property $x=\frac{243}{3}$      Divide by 3 x = 81        Evaluate

 Related Links: Math algebra Logarithmic Equations: Natural Base - Simple Equations Algebra Topics Logarithmic Equations

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