Question

Verify the Identity. $$\log 10^{\operatorname{tan} t}=\tan t$$

   Verify the Identity.
$$\log 10^{\operatorname{tan} t}=\tan t$$

Precalculus: Functions and Graphs

Precalculus: Functions and Graphs

Earl W. Swokowski,… 11th Edition

Chapter 6, Problem 45

Instant Answer

Solved by Expert Babita Kumari

12/07/2020

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Step 1: We start with the left-hand side of the equation, which is $\log 10^{\operatorname{tan} t}$. Show more…

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Verify the Identity. $$\log 10^{\operatorname{tan} t}=\tan t$$

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Key Concepts

Fundamental Logarithm Property

A key property of logarithms is that the logarithm of a number raised to an exponent, when the logarithm's base matches the number's base, simplifies directly to the exponent; mathematically, log_b(b^x) = x. This property is essential in verifying identities such as log 10^(tan t) = tan t, allowing one to cancel the base and reveal the exponent.

Logarithm Function

The logarithm function is the inverse of exponentiation. It translates the operation of raising a number to a power into multiplication, and vice versa. This property forms the basis of many algebraic simplifications and identity verifications by allowing conversion between logarithmic and exponential forms.

Exponential Function

An exponential function involves raising a constant base to a variable exponent. Its critical relationship with the logarithm function, as inverses, allows for straightforward conversion between exponential expressions and logarithms. This relationship is often used to simplify expressions and solve equations involving exponentials and logarithms.

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