Solving Logarithmic Equations(Day 1)

Warm-Up

• Factor: 𝑥𝑥2 − 3𝑥𝑥 − 40

Product = -40, Sum = -3… factors are -8 and 5 (both over 1)

So (x – 8)(x + 5) is the factored form

Essential Question

• How can I use exponential form to help me solve logarithmic equations?

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 to this 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”, try rewriting the problem in exponential form.

Conversely, when working with exponential expressions, if ever you get “stuck”, try rewriting the problem in logarithmic form.

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

62 = xWe’re finished, x = 36

6Solve for x: log 2x =Example 1

Solution:

5y =125

Rewrite the problem in exponential form.

Since 125

= 5− 2

5y = 5− 2y = −2

51Solve for y: log25

y=Example 2Alternate: If you can evaluate the log, you will already have the value of the y!

y = log5 (1/52) = log5 5-2 = -2

Example 3

Solution: Rewrite in exponential form.

Solve for x… logx 54 = 4x4 = 54

Now use the 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 + 4

Simplify 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) = 2log2 (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.

Suppose b > 0 and b ≠ 1.Then logb x1 = logb x2 if and only if x1 = x2

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

Example 7 Solve: log3 (4x +10) = log3 (x +1)

Solution: Since the bases are both ‘3’ we simply set the arguments equal to each other and solve.

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!!! … log3(−2)The log of a negative number does not exist.

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

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 = 5x

x2 − 5x −14 = 0(x − 7)(x + 2) = 0Factor the quadratic

and solve(x − 7) = 0 or (x + 2) = 0

x = 7 or x = −2continued on the next page…

Example 8 (cont) Solve: log8 (x2 −14) = log8 (5x)

Solution: x = 7 or x = −2It 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 because it causes the log of a negative #.

…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… log8 35 on both sides!

x = 7 only

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 = xx = 8

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

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 the fraction.

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

Assignment:

WS: Solving Log Equations

(Day 2 Practice)