3. The Logarithm Laws

by M. Bourne

Since a logarithm is simply an exponent which is just being written down on the line, we expect the logarithm laws to work the same as the rules for exponents, and luckily, they do.

Exponents
Logarithms

bm × bn = bm+n
logb xy = logb x + logb y

bm ÷ bn = bm-n
logb (x/y) = logb x − logb y

(bm)n = bmn
logb (xn) = n logb x

b1 = b
logb (b) = 1

b0 = 1
logb (1) = 0


Examples

In these examples, I am not including any base, since the laws hold for logarithms with any positive integer base.

1. Expand log 7x as the sum of 2 logarithms.

Answer


2. Using your calculator, show that log (20/5) = log 20 − log 5.

(On our calculators, "log" is actually "log base 10", or log10.)

Answer


3. Express as a multiple of logarithms: log x5.

Answer


4. All of these are equal to 1:

log6 6 = log10 10 = logx x = loga a = 1


5. All of these are equivalent to 0:

log7 1 = log10 1 = loge1 = logx 1 = 0



Exercises

1. Express as a sum, difference, or multiple of logarithms:

math expression

Answer


2. Express

math expression

as the logarithm of a single quantity.

Answer


3. Determine the exact value of:

math expression

Answer


4. Solve for y in terms of x: log2x + log2y = 1

Answer



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