/*! 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 9 Solve for \(x\). $$ \log _{7... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 9

Solve for \(x\). $$ \log _{7} 49=x $$

Short Answer

Expert verified
\( x = 2 \)

Step by step solution

01

Understand the Logarithmic Equation

The given problem is \( \log_{7} 49 = x \). This reads as "the logarithm of 49 to the base 7 equals \( x \)". Our goal is to find the value of \( x \). It means finding the power to which 7 must be raised to get 49.
02

Rewrite the Logarithm in Exponential Form

Recall the definition of a logarithm: if \( \log_{b}(a) = c \), then \( b^{c} = a \). Applying this to our problem: \( 7^{x} = 49 \). Now, we need to find \( x \) such that this equation holds true.
03

Express 49 as a Power of 7

Recognize that 49 is a power of 7. In fact, \( 49 = 7^{2} \). Rewrite the equation from Step 2 as \( 7^{x} = 7^{2} \).
04

Solve for \(x\)

When two bases are the same, their exponents must be equal. Therefore, from the equation \( 7^{x} = 7^{2} \), we can equate the exponents: \( x = 2 \).

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.

Exponential Form
Understanding exponential form is crucial when working with logarithmic equations. Exponential form helps us convert a logarithmic expression into a more familiar equation. In general, if you have a logarithmic equation like \( \log_{b}(a) = c \), the exponential form would be \( b^{c} = a \). This transformation is vital as it makes the equation simpler and easier to solve.
For example, when we dealt with the equation \( \log_{7}(49) = x \), we converted it to its exponential form: \( 7^{x} = 49 \).
  • \( b \) is the base of the logarithm and becomes the base of the exponential expression.
  • \( c \) is the logarithm result and becomes the exponent.
  • \( a \) is the result of the power operation.
By rewriting logarithmic forms in their exponential equivalents, we simplify the task of finding the unknown values in equations.
Logarithm Properties
Logarithms come with a set of properties that make them incredibly powerful tools in mathematics.
These properties include the product property, quotient property, and power property. Each of these properties helps us manipulate and transform logarithmic expressions.
For solving equations, you often rely on these basic properties:
  • Product Property: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \).
  • Quotient Property: \( \log_{b}(\frac{m}{n}) = \log_{b}(m) - \log_{b}(n) \).
  • Power Property: \( \log_{b}(m^{n}) = n\log_{b}(m) \).
These properties simplify complex logarithmic operations and allow us to break down complicated problems into more manageable steps.
In our original exercise, recognizing that \( 49 = 7^{2} \) allowed us to make use of the properties of exponents.
Base and Exponent
At the heart of both logarithmic and exponential expressions are the notions of base and exponent.
Understanding their roles is vital for solving problems effectively.
  • Base: In the logarithmic expression \( \log_{b}(a) = c \), \( b \) represents the base. It is the same in the exponential form, governing the power operation.
  • Exponent: The term \( c \) is the exponent in the exponential equation and represents the power to which the base must be raised to yield \( a \).
The exponent tells you how many times to use the base as a factor.
In our original exercise with \( \log_{7}(49) = x \), the base is 7, and by expressing 49 as \( 7^{2} \), it was clear that the exponent \( x \) should also be 2.
Grasping the relationship between base and exponent allows you to identify potential patterns and solutions more quickly.

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

Use the Tangent feature from the DRAW menu to find the rate of change in part (b). Perriot's Restaurant purchased kitchen equipment on January 1,2014 . The value of the equipment decreases by \(15 \%\) every year. On January \(1,2016,\) the value was \(\$ 14,450 .\) a) Find an exponential model for the value, \(V,\) of the equipment, in dollars, \(t\) years after January 1 \(2016 .\) b) What is the rate of change in the value of the equipment on January \(1,2016 ?\) c) What was the original value of the equipment on January \(1,2014 ?\) d) How many years after January 1,2014 will the value of the equipment have decreased by half?

The value, \(V(t),\) in dollars, of a share of Cypress Mills stock \(t\) months after it is purchased is modeled by $$ V(t)=58\left(1-e^{-1.1 t}\right)+20$$. a) Find \(V(1)\) and \(V(12)\). b) Find \(V^{\prime}(t)\). c) After how many months will the value of a share of the stock first reach \(\$ 75 ?\) d) Find \(\lim _{t \rightarrow \infty} V(t)\). Discuss the value of a share over a long period of time. Is this trend typical?

A beam of light enters a medium such as water or smoky air with initial intensity \(I_{0}\). Its intensity is decreased depending on the thickness (or concentration) of the medium. The intensity I at a depth (or concentration) of \(x\) units is given by $$I=I_{0} e^{-\mu x}$$ The constant \(\mu\left({ }^{u} m u "\right),\) called the coefficient of absorption, varies with the medium. Use this law for Exercises 62 and \(63 .\) Sea water has \(\mu=1.4\) and \(x\) is measured in meters. What would increase cloudiness more - dropping \(x\) from \(2 \mathrm{~m}\) to \(5 \mathrm{~m}\) or dropping \(x\) from \(7 \mathrm{~m}\) to \(10 \mathrm{~m}\) ? Explain.

Find the minimum value of \(f(x)=x e^{x}\) over [-2,0]

We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. The rapidly growing sales of organic foods
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied 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.