Laws of Logarithms

5-6: Laws of Logarithms

1) Log_b MN = Log_b M + Log_bN

2) Log_b M/N = Log_b M - Log_bN

3) Log_b M = Log_b N if and only if M = N

4) Log_b M^k = k Log_bM

5) Log_b b = 1

6) Log_b 1 = 0

7) Log_b b^k = k
8) _bLog_bx _{= x}

Sample problems

Write each log in expanded form.

1) Log₅ xy² =

Solution:

Log₅ x + Log₅ y²
= Log₅ x + 2 Log₅ y



2) Log₇(xy/z²) =

Solution:

Log₇ x + Log₇ y - 2 Log₇ z

3)

Express each as a single log.

1) Log x + Log y - Log z =

Solution:

Log (xy)/z

2) 2 Ln x + 3 Ln y =

Solution:

Ln x²y³

3) (1/2) Ln x - (1/3) Ln y =

Solution:

Writing logs as single logs can be helpful in solving many log equations.

1) Log₂(x + 1) + Log₂ 3 = 4

Solution:

First combine the logs as a single log.

Log₂ 3(x + 1) = 4

Now rewrite as an exponential equation.
3(x + 1) = 2⁴

Now solve for x.

3x + 3 = 16

3x = 13

x = 13/3 Since this doesn't make the number inside the log zero or negative, the answer is acceptable.

2) Log (x + 3) + Log x = 1

Solution:

Again, combine the logs as a single log.

Log x(x + 3) = 1

Rewrite as an exponential.

x(x + 3) = 10

Solve for x.

x² + 3x = 10

x² + 3x - 10 = 0

(x + 5)(x - 2) = 0

x = -5 or x = 2 We have to throw out 5. Why? Because it makes (x + 3) negative and we can't take the log of a negative number. So the only answer is x = 2.



3) Ln (x - 4) + Ln x = Ln 21

Solution:

Notice, this time we have a log on both sides. If we write the left side as a single log, we can use the rule that if the logs are equal, the quantity inside must be equal.

Ln x(x - 4) = Ln 21

Since the logs are equal, what is inside must be equal.

x(x - 4) = 21

Solve for x.

x² - 4x = 21

x² - 4x - 21 = 0

(x - 7)(x + 3) = 0

x = 7 or x = -3 Again, we need to throw out one of the answers because it makes both quantities negative. Throw out -3 and keep 7. Thus, the answer is x = 7.

Simplify each log

1) ln e⁵

Solution:

This is rule number 7. The answer is 5!

2) Log 10^-3

Solution:

This is again rule #7. The answer: -3 (This answers the question: what power do you raise 10 to get 10 to the third?

3) e^{ln 7}

Solution:

This is rule #8. The answer is 7.

4) e^{2ln 5}

Solution:

We can use rule #8 as soon as we simplify the problem.
Rewrite as: e^{ln 25}
= 25 The 25 came from 5².

5) 10^{Log 6}

Solution:

Rule #8 again. Answer: 6

6) 10^{2 + log 5}

Solution:

We need to simplify before we can apply one of the rules.
Rewrite as: (10²)(10^log5) Adding exponents means you are multiplying the bases.

= 100(5) Use rule #8 again.

= 500