/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 120 Using Properties of Exponents Gi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 120

Using Properties of Exponents Given the exponential function \(f(x)=a^{x},\) show that (a) \(f(u+v)=f(u) \cdot f(v)\) (b) \(f(2 x)=[f(x)]^{2}\)

Short Answer

Expert verified
Through the properties of exponents, we can show that \(f(u+v)=f(u) \cdot f(v)\) and \(f(2x)=[f(x)]^{2}\). This is because any base raised to the sum of two exponents can be written as the product of that base (same base) raised to each of the exponents individually. Also, the base raised to a power times another power can be written as the base raised to the product of the two powers. These properties when applied to the function \(f(x) = a^{x}\) prove both (a) and (b).

Step by step solution

01

Use the properties of exponents to simplify \(f(u+v)\)

The equation given is \(f(u+v)=a^{u+v}\). According to the property of exponents, this can be simplifed as \(a^{u} \cdot a^{v}\). Remembering the function, this would be the same as \(f(u) \cdot f(v)\). So \(f(u+v)=f(u) \cdot f(v)\).
02

Use the properties of exponents to simplify \(f(2x)\)

The second equation is \(f(2x)=a^{2x}\). This can be simplified as \([a^{x}]^{2}\), which can also be written as \([f(x)]^{2}\). So \(f(2x)=[f(x)]^{2}\), thus proving the second part of the problem.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

$$ \int \frac{\sqrt{x}}{\sqrt{1+x^{3}}} d x $$

Use implicit differentiation to find an equation of the tangent line to the graph of the equation at the given point. \(\arctan (x y)=\arcsin (x+y), \quad(0,0)\)

Using the Area of a Region Find the value of \(a\) such that the area bounded by \(y=e^{-x},\) the \(x\) -axis, \(x=-a,\) and \(x=a\) is \(\frac{8}{3} .\)

In Exercises 106鈥�108, verify the differentiation formula. $$ \frac{d}{d x}\left[\sinh ^{-1} x\right]=\frac{1}{\sqrt{x^{2}+1}} $$

Analyzing a Graph Consider the function $$ f(x)=\frac{2}{1+e^{1 / x}} $$ (a) Use a graphing utility to graph \(f\) (b) Write a short paragraph explaining why the graph has a horizontal asymptote at \(y=1\) and why the function has a nonremovable discontinuity at \(x=0\) .
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.