We now know that a logarithm is perhaps best understood as being closely related to an exponential equation.

In fact, whenever we get stuck in the problems that follow we will return tothis one simple insight.

We might even state it as a simple rule.

We now know that a logarithm is perhaps best understood as being closely related to an exponential equation.

In fact, whenever we get stuck in the problems that follow we will return tothis one simple insight.

We might even state it as a simple rule.

Solving equations with logarithms:

When working with logarithms,if ever you get “stuck”, tryrewriting the problem inexponential form.

When working with logarithms,if ever you get “stuck”, tryrewriting the problem inexponential form.

Conversely, when workingwith exponential expressions,if ever you get “stuck”, tryrewriting the problemin logarithmic form.

Conversely, when workingwith exponential expressions,if ever you get “stuck”, tryrewriting the problemin logarithmic form.

Solution:Let’s rewrite the problem in exponential form.62 x

We’re finished, x = 36

6Solve for x: log 2x

Example 1

Solution:

5y 1

25

Rewrite the problem in exponential form.

Since 1

255 2

5y 5 2

y 2

5

1Solve for y: log

25y

Example 2

Example 3

Try setting this up like this:

Solution:

Rewrite in exponential form.

Solve for x… logx 54 = 4

x4 = 54

Now use your reciprocal power to solve.

(x4 )1/4= (54)1/4

x = (54)1/4 So x ≈ 2.7108

Example 4

Solution:

Rewrite in exponential form.

Solve for x… log3 (x+4) = 4

34 = x + 4Simplify the power expression.

81 = x + 4

x = 77Subtract 4 to finish…

Example 5: More complicated expressions

KEY KNOWLEDGE: You ALWAYS have to isolate the log expression before you change forms.

Rewrite in exponential form.Remember common log is base 10.

Solve for x… 4 log (x+1) = 8

102 = x + 1

Evaluate the power. 100 = x + 1

x = 99Subtract 1 to finish…

Divide by the 4… log (x+1) = 2

Example 6: More complicated expressions

KEY KNOWLEDGE: You ALWAYS have to isolate the log expression before you change forms.

Rewrite in exponential form..

Solve for x… 0.5 log2 (3x+1) – 6 = –4

24 = 3x + 1Evaluate the power. 16 = 3x + 1

x = 5Subtract 1, then divide by 3 to finish…

Add 6… 0.5 log2 (3x+1) = 2 log2 (3x+1) = 4Multiply by 2 (or divide by 0.5)

15 = 3x

Finally, we want to take a look at the Property of Equality for Logarithmic Functions.

Finally, we want to take a look at the Property of Equality for Logarithmic Functions.

Suppose b 0 and b 1.

Then logb x1 logb x2 if and only if x1 x2

Basically, with logarithmic functions,if the bases match on both sides of the equal sign , then simply set the arguments equal.

Basically, with logarithmic functions,if the bases match on both sides of the equal sign , then simply set the arguments equal.

Example 7

Solve: log3 (4x 10) log3 (x 1)

Solution:Since the bases are both ‘3’ we simply set the arguments equal.

4x 10 x 13x 10 1

3x 9x 3

Is the –3 a valid solution or is it extraneous??

Note: If I plug -3 into the original, it causes me to take the log of a negative number!!!

Therefore, it is NOT valid. Our answer is NO SOLUTION…

Example 8

Solve: log8 (x2 14) log8 (5x)

Solution:Since the bases are both ‘8’ we simply set the arguments equal.x2 14 5xx2 5x 14 0(x 7)(x 2) 0

Factor the quadratic and solve

(x 7) 0 or (x 2) 0x 7 or x 2 continued on the next

page

Example 8 (cont)

Solve: log8 (x2 14) log8 (5x)

Solution:x 7 or x 2

It appears that we have 2 solutions here.If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive bases, but also we see that the arguments must be positive as well. Therefore, -2 is not a solution. Each side becomes the log8 (-10) !!!!! So that answer is extraneous and has to be thrown out. Only x = 7 works in the original equation.

Example 9: Equations with more than one log (Using our Condensing Skills)

KEY KNOWLEDGE: You ALWAYS have to isolate the log expression before you change forms. You may have to CONDENSE it!

Now solve... Subtract 3x .

Solve for x… log2 (3x + 8) = log2 4 + log2 x

8 = x x = 8

Condense the right side … log2 (3x+8) = log2 (4x) (Product Property)

3x + 8 = 4xProperty of Equality…

Example 10: Equations with more than one log (Using our Condensing Skills)

KEY KNOWLEDGE: You ALWAYS have to isolate the log expression before you change forms. You may have to CONDENSE it!

Now solve... Divide by 24 and simplify .

Solve for x… log2 (3x) + log2 (8) = 4

x = 16/24 x = 2/3

Condense the left side … log2 (24x) = 4 (Product Property… 3x*8 = 24x)

24 = 24xChange into exponential form…

Evaluate the power…16 = 24x

HOMEWORK:

• Regular Algebra 2… WS – Solving Log Equations

• Pre-AP Algebra 2… [Prentice Hall book]p. 472 (33 – 47) Odd, (82 & 92)