Skip to main content
Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 1.75
Chapter 1, Problem 1.75

Convert the following expressions to the indicated base.


a1lnaa^{\(\frac{1}{\ln a}\)} using basa e, for a>0a > 0 and a≠1a ≠ 1

Verified step by step guidance
1
Start by recognizing that the expression \( a^{\frac{1}{\ln a}} \) can be rewritten using the property of exponents and logarithms. We know that \( a^x = e^{x \ln a} \).
Apply this property to the given expression: \( a^{\frac{1}{\ln a}} = e^{\frac{1}{\ln a} \cdot \ln a} \).
Simplify the exponent: \( \frac{1}{\ln a} \cdot \ln a = 1 \).
Substitute back into the expression: \( e^{1} \).
Conclude that the expression simplifies to \( e \).>

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Functions

Exponential functions are mathematical expressions in the form of f(x) = a^x, where 'a' is a positive constant. These functions are characterized by their rapid growth or decay, depending on the base. Understanding how to manipulate and convert between different bases is crucial for solving problems involving exponential expressions.
Recommended video:
6:13
Exponential Functions

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is the inverse function of the exponential function with base 'e'. The natural logarithm is essential for converting exponential expressions into a more manageable form, particularly when dealing with expressions like a^(1/ln(a)).
Recommended video:
05:18
Derivative of the Natural Logarithmic Function

Change of Base Formula

The change of base formula allows for the conversion of logarithms from one base to another. It states that log_b(a) = log_k(a) / log_k(b) for any positive k. This concept is particularly useful when converting expressions to a specific base, such as converting a^x to base 'e' in the given problem.
Recommended video:
05:36
Change of Base Property
Related Practice
Textbook Question

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = 4 - 4x + x²

407
views
Textbook Question

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = (1/x) - x²

260
views
Textbook Question

Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.


cos⁻¹ √3/2

330
views
Textbook Question

Solve each equation.

48=6e4k48=6e^{4k}

324
views
Textbook Question

Working with composite functions Find possible choices for outer and inner functions ƒ and g such that the given function  h equals ƒ o g .


h(x) = √ (x⁴ + 2 )

439
views
Textbook Question

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = x⁴

273
views