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More Logarithms and Algebra Lessons

In these lessons, we will look at how to evaluate simple logarithmic functions and solve for *x* in logarithmic functions.

The following diagram shows how logarithm and exponents are related. Scroll down the page for examples and solutions.

Take note of the following:

- Logarithms of a number to the base of the same number is 1, i.e. log
_{a}*a*= 1 - Logarithms of 1 to any base is 0, i.e. log
1 = 0_{a} - Log
0 is undefined_{a} - Logarithms of negative numbers are undefined.
- The base of logarithms cannot be negative or 1.

* Example: *

Calculate the value of each of the following:

a) 1og_{2} 64

b) log_{9} 3

c) log_{4} 1

d) log_{6} 6

e) log_{8} 0.25

f) log_{3}–9

* Solution:*

a) Let *x* = log_{2} 64

2* ^{x} * = 64

b) Let *x* = log_{9} 3

9* ^{x} * = 3

c) Let *x* = log_{4} 1

4* ^{x}* = 1

d) Let *x* = log_{6} 6

6* ^{x}* = 6

e) Let *x = *log_{8} 0.25

8* ^{x}* = 0.25

f) Let *x* = log_{3}– 9

3* ^{x}* = – 9

**Example**

Solve log* _{x}* 4 = 2

**Solution:**

log* _{x}* 4 = 2

Since *x* is the base, *x* > 0 and *x* ≠ 1; so *x* = –2 is rejected and the only solution is *x* = 2

**Example:**

Solve log_{ 3} *x = *2

**Solution:**

log _{3} *x = *2

3^{2} = *x
*

**Example:**

Solve log * _{x}* (4

**Solution:**

log * _{x}* (4

(

So,

Just as we can use logarithms to access exponents in exponential equations, we can use exponentiation to access the insides of a logarithm. Solving logarithmic equations often involves exponentiating logarithms in order to get rid of the log and access its insides. Sometimes we can use the product rule, the quotient rule, or the power rule of logarithms to help us with solving logarithmic equations.

This video shows how solve a logarithmic equation using properties of logarithms and some other algebra techniques.

Example: Solve 2log

When given a problem on solving a logarithmic equation with multiple logs, students should understand how to condense logarithms. By condensing the logarithms, we can create an equation with only one log, and can use methods of exponentiation for solving a logarithmic equation with multiple logs. This requires knowledge of the product, quotient and power rules of logarithms.

Example: Solve log

Examples: Solve and identify any extraneous solutions

a) log

b) log

c) log(x + 2) + log(x -1) = 1

d) log x

Examples:

log

log(5x - 1) = 2 + log(x - 2)

ln x = 1/2 ln(2x + 5/2) + 1/2 ln 2

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