<< Chapter < Page | Chapter >> Page > |
Write the following exponential equations in logarithmic form.
Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider log28. We ask, “To what exponent must 2 be raised in order to get 8?” Because we already know 23=8, it follows that log28=3.
Now consider solving log749 and log327 mentally.
Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate log2349 mentally.
Given a logarithm of the form y=logb(x), evaluate it mentally.
Solve y=log4(64) without using a calculator.
First we rewrite the logarithm in exponential form: 4y=64. Next, we ask, “To what exponent must 4 be raised in order to get 64?”
We know
Therefore,
Solve y=log121(11) without using a calculator.
log121(11)=12 (recalling that √121=(121)12=11 )
Evaluate y=log3(127) without using a calculator.
First we rewrite the logarithm in exponential form: 3y=127. Next, we ask, “To what exponent must 3 be raised in order to get 127? ”
We know 33=27, but what must we do to get the reciprocal, 127? Recall from working with exponents that b−a=1ba. We use this information to write
Therefore, log3(127)=−3.
Evaluate y=log2(132) without using a calculator.
log2(132)=−5
Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression log(x) means log10(x). We call a base-10 logarithm a common logarithm . Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.
A common logarithm is a logarithm with base 10. We write log10(x) simply as log(x). The common logarithm of a positive number x satisfies the following definition.
For x>0,
We read log(x) as, “the logarithm with base 10 of x ” or “log base 10 of x. ”
The logarithm y is the exponent to which 10 must be raised to get x.
Given a common logarithm of the form y=log(x), evaluate it mentally.
Notification Switch
Would you like to follow the 'Precalculus' conversation and receive update notifications?