/*! 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 7 Write each equation in its equiv... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 7

Write each equation in its equivalent exponential form. $$ \log _{6} 216=y $$

Short Answer

Expert verified
The equivalent exponential form of \(\log _{6} 216 = y\) is \(6^y = 216\).

Step by step solution

01

Identify the base, exponent and result

In the logarithm \(\log _{6} 216=y\), the base of the logarithm is 6, the result of the logarithm is 216, and the variable y is equivalent to the exponent in the exponential form.
02

Apply Logarithmic to Exponential Conversion

Now that we have identified the base, result, and exponent, we can convert the logarithm to its equivalent exponential form using the property \(b^c = a\). Equating the terms we have identified to the conversion property gives us: \(6^y = 216\). This is the exponential form of \(\log _{6} 216=y\).

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.

Logarithmic Form
Logarithms are a unique way to express exponentiation in reverse. When we see a logarithm like \(\log _{6} 216 = y\), it tells us the exponent \(y\) to which the base \(6\) must be raised to get the number \(216\). In simple terms, a logarithm answers the question: "To what power must the base be raised to produce a given value?"

Logarithmic form is expressed as \(\log_b a = y\), where:
  • \(b\) is the base of the logarithm
  • \(a\) is the result or the number for which we are solving the logarithm
  • \(y\) is the power or exponent that the base is raised to, resulting in \(a\)
Understanding logarithmic form helps in converting it to exponential form for easier calculation or comprehension.
Logarithm Properties
Understanding the properties of logarithms is crucial for simplifying and manipulating logarithmic expressions. Here are some key properties:

  • Product Property: \(\log_b (mn) = \log_b m + \log_b n\), which combines the log of a product into the sum of logs.
  • Quotient Property: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\), expressing the log of a quotient as a difference of logs.
  • Power Property: \(\log_b (m^n) = n \cdot \log_b m\), which moves the exponent in the argument to the front as a multiplier.
  • Change of Base Formula: \(\log_b a = \frac{\log_k a}{\log_k b}\), which allows conversion of logs to a different base \(k\).
These properties not only simplify complex logarithmic expressions but also aid in understanding their structure and behavior. Any logarithm expression can be manipulated using these properties to make computations easier.
Base of Logarithm
The base of a logarithm is a fundamental aspect because it determines the exponential growth factor. When we see an expression like \(\log_b a\), \(b\) is the base and it signifies the number that is repeatedly multiplied.

The base is crucial in conversions and calculations within both exponential and logarithmic forms:
  • If \(b = 10\), it's a common logarithm, widely used in scientific and engineering contexts.
  • If \(b = e\) (approximately 2.718), it's a natural logarithm, denoted as \(\ln\), prevalent in calculus and natural growth models.
In our problem, the base is \(6\), which means answering "to what power must 6 be raised to get 216?" This idea helps when switching between exponential and logarithmic formats and is essential in solving problems efficiently.

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

The function \(W(t)=2600\left(1-0.51 e^{-0.075 t}\right)^{3}\) models the weight, \(W(t),\) in kilograms, of a female African elephant at age \(t\) years. (1 kilogram \(=2.2\) pounds) Use a graphing utility to graph the function. Then TRACE along the curve to estimate the age of an adult female elephant weighing 1800 kilograms.

Evaluate or simplify each expression without using a calculator. $$ \ln \frac{1}{e^{6}} $$

Solve each equation. Check each proposed solution by direct substitution or with a graphing utility. $$ (\ln x)^{2}=\ln x^{2} $$

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the function to solve. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?

Find the domain of each logarithmic function. $$ f(x)=\log _{5}(x+6) $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

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.