Question

Use the property: $b^{a}=c$ if and only if $\log _{b}(c)=a$ from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. $\log (0.1)=-1$

Name: Problem 13, Chapter 6: Precalculus
Uploaded: 2021-08-14T14:13:04-08:00
Duration: 30 s
Channel: Nick Johnson
Description: Use the property: b^a=c if and only if logb(c)=a from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. log(0.1)=-1

   Use the property: $b^{a}=c$ if and only if $\log _{b}(c)=a$ from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations.
$\log (0.1)=-1$

Precalculus

Carl Stitz, Jeff… 3rd Edition

Chapter 6, Problem 13

Instant Answer

Solved by Expert Nick Johnson

08/14/2021

Step 1

It is $\log (0.1)=-1$. Here, the base of the logarithm is not mentioned, so by default, it is 10. So, the equation is $\log _{10}(0.1)=-1$. Show more…

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Use the property: $b^{a}=c$ if and only if $\log _{b}(c)=a$ from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. $\log (0.1)=-1$

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Key Concepts

Conversion between Exponential and Logarithmic Forms

This concept involves the equivalence between the two forms: exponential and logarithmic. Specifically, an equation expressed as b^a = c can be rewritten as log_b(c) = a, and conversely, a logarithmic equation can be expressed in its exponential form. This conversion is fundamental in understanding and solving equations that involve exponents and logarithms.

Inverse Relationship of Exponentiation and Logarithms

This key concept highlights that logarithms are the inverse operations of exponentiation. Recognizing that applying a logarithm undoes the effect of raising a number to a power allows one to manipulate and solve equations by changing their form. This understanding is critical when converting between forms, as it directly applies the principle that an operation and its inverse will yield the original value.

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