What is a Logarithm?

MATH REVIEW: USEFUL MATH FOR EVERYONE

SECTION 4.5. WHAT IS A LOGARITHM?

7. 10^{(log
a)} = a (or, in the case of natural logarithms, e^{(ln
a)} = a). Logarithms and exponents reverse each other.

For example:

10^{(log
3)} = 3

10^{(log
8)} = 8

e^{(ln
3)} = 3

e^{(ln
8)} = 8

If you raise a number to the power of a logarithm that has that number as its base, it is equal to the number that you used in the logarithm.

8. log (10^r) = r (in the case of natural logarithms, ln e^r = r)Because logarithms and exponents reverse each other, this rule is similar to rule number seven.

For example:

log (10²) = 2

log (10³) = 3

ln (e²) = 2

ln (e⁴) = 4

Any logarithm of its base number raised to some exponent is equal to that exponent.

9. log (1/a) = -log a means that the logarithm of 1 divided by some number is equal to the negative logarithm of that number. (This is the exactly the opposite of the rule governing exponents where a number raised to a negative number is equal to 1 divided by that number raised to that power.)

For example:

log (1/2) = - log 2 = -0.301

log (1/3) = - log 3 = -0.477

ln (1/2) = -ln 2 = -0.693

ln (1/3) = -ln 3 = -1.099

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