Math  /  Algebra

QuestionRewrite the following in exponential form: a) log8(18)=1\log _{8}\left(\frac{1}{8}\right)=-1 \square b) log(117)=x\log (117)=x \square c) ln(x)=1\ln (x)=-1 \square

Studdy Solution

STEP 1

What is this asking? We need to rewrite logarithmic expressions into their equivalent exponential forms. Watch out! Remember the relationship between logs and exponents: logb(a)=c\log_{b}(a) = c means bc=ab^c = a.
Don't mix up the base, exponent, and result!

STEP 2

1. Rewrite Logarithm with Base 8
2. Rewrite Logarithm with Base 10
3. Rewrite Natural Logarithm

STEP 3

We're given log8(18)=1\log_{8} \left( \frac{1}{8} \right) = -1.
Our **base** is 88, our **exponent** is 1-1, and our **result** is 18\frac{1}{8}.

STEP 4

Rewriting this in exponential form gives us 81=188^{-1} = \frac{1}{8}.
Remember, a **negative exponent** means we take the **reciprocal**!

STEP 5

We have log(117)=x\log(117) = x.
When the **base** isn't written, it's understood to be 1010!
So, our **base** is 1010, our **exponent** is xx, and our **result** is 117117.

STEP 6

In exponential form, this becomes 10x=11710^x = 117.
See how the **logarithm** tells us what **exponent** we need to raise 1010 to in order to get 117117?

STEP 7

We're given ln(x)=1\ln(x) = -1.
Remember, ln\ln is just a shorthand for loge\log_e, where ee is the **magical mathematical constant** approximately equal to 2.7182.718.
So, our **base** is ee, our **exponent** is 1-1, and our **result** is xx.

STEP 8

Rewriting in exponential form gives us e1=xe^{-1} = x, or x=1ex = \frac{1}{e} because of that **negative exponent** action!

STEP 9

a) 81=188^{-1} = \frac{1}{8} b) 10x=11710^x = 117 c) e1=xe^{-1} = x

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