/*! 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 21 Solve the given problem for \(X\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 21

Solve the given problem for \(X\). $$ 6^{x+\pi}=2 $$

Short Answer

Expert verified
\(x = \frac{\ln(2)}{\ln(6)} - \pi\)

Step by step solution

01

Take the Logarithm

To solve for \(x\), take the natural logarithm (ln) of both sides of the equation \(6^{x+\pi} = 2\). This gives us \(\ln(6^{x+\pi}) = \ln(2)\).
02

Apply the Power Rule for Logarithms

Using the power rule for logarithms, which states \(\ln(a^b) = b \cdot \ln(a)\), rewrite the left side: \((x+\pi) \cdot \ln(6) = \ln(2)\).
03

Isolate \(x\)

Divide both sides by \(\ln(6)\) to solve for \(x + \pi\): \(x + \pi = \frac{\ln(2)}{\ln(6)}\).
04

Solve for \(x\)

Subtract \(\pi\) from both sides to isolate \(x\): \(x = \frac{\ln(2)}{\ln(6)} - \pi\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Logarithms
Logarithms are the inverse operation to exponentiation. They help to solve equations where the unknown variable is an exponent. When you see an equation like \( 6^{x+\pi} = 2 \), the goal is to find the exponent \( x \). To "undo" the exponentiation, we take the logarithm on both sides of the equation.
This allows us to bring down the exponent as a coefficient, simplifying the equation. Taking the logarithm of both sides is a crucial first step in solving exponential equations.
The Natural Logarithm (\(\ln\))
The natural logarithm, denoted as \(\ln\), is a specific kind of logarithm that uses the base \( e \), where \( e \approx 2.71828 \). It is often used because of its natural properties in calculus and its ease in simplifying exponential equations.
Using the natural logarithm in our exercise helps convert the exponential form \( 6^{x+\pi} = 2 \) to \( \ln(6^{x+\pi}) = \ln(2) \). This provides a straightforward way to handle the exponent when solving for \( x \).
Power Rule for Logarithms
The power rule for logarithms is a handy tool that allows us to work with exponents. It states that \( \ln(a^b) = b \cdot \ln(a) \). This means you can take an exponent down and multiply it by the logarithm of the base.
In our equation \( \ln(6^{x+\pi}) = \ln(2) \), we apply the power rule so it becomes \( (x+\pi) \cdot \ln(6) = \ln(2) \). This turns the problem into a linear equation where isolating the variable is much simpler.
Solving for Variables
Solving for variables involves isolating the variable of interest on one side of the equation. Once we've rewritten \( (x+\pi) \cdot \ln(6) = \ln(2) \), our goal is to isolate \( x \).
We divide both sides by \( \ln(6) \) to isolate \( x + \pi \), giving us \( x + \pi = \frac{\ln(2)}{\ln(6)} \). Finally, we subtract \( \pi \) from both sides, resulting in \( x = \frac{\ln(2)}{\ln(6)} - \pi \). Each step in this process simplifies the problem and brings us closer to the solution.

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

What questions are you answering when you evaluate \(\log _{5}(25) ?\)

In exercises \(25-27,\) find all real roots of the given function. $$ f(x)=x^{3}+x^{2}-x $$

In exercises \(11-17,\) simplify the given term and write your answer without negative exponents. $$ \left(\frac{-3 s^{2 / 3} t^{2}}{4 s^{3} t^{5 / 3}}\right)^{3} $$

Simplify the given expressions. $$ -2(11-5) \div 3+2^{3} $$

In exercises \(5-7,\) write the given term without using exponents. $$ (-2 x+y)^{-1 / 5} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.