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Chapter 5: Problem 48

Convert to an exponential equation. \(\log 7=0.845\)

Short Answer

Expert verified
The exponential form is \(10^{0.845} = 7\).

Step by step solution

01

Understand the Logarithmic Equation

The given logarithmic equation is \(\text{log} 7 = 0.845\). A logarithmic equation \(\text{log}_b a = c\) can be converted to an exponential form by using the definition of logarithms.
02

Identify the Components

In the equation \(\text{log} 7 = 0.845\), the base of the logarithm can be assumed to be 10 as it is not specified. Therefore, this can be rearranged using the form \(\text{log}_{10} 7 = 0.845\).
03

Convert to Exponential Form

Using the definition of logarithms, \(\text{log}_b a = c \) can be rewritten in exponential form as \(b^c = a\). Substituting the values, we get \(10^{0.845} = 7\).

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Key Concepts

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

Logarithmic Equations
Logarithmic equations are mathematical expressions that involve logarithms, which are the inverse operations of exponentiation. In simpler terms, if you have an equation like \(\text{log}_b a = c\), it means finding the exponent 'c' to which the base 'b' must be raised to get the number 'a'.
For example, \(\text{log}_{10} 100 = 2\) because raising 10 to the power of 2 gives 100.
Conversion between forms
One of the most fundamental skills when working with logarithmic and exponential equations is converting between their forms. The logarithmic form \(\text{log}_b a = c\) can be converted to the exponential form \(\text{b}^c = a\).
Let's break it down with the given exercise: \(\text{log} 7 = 0.845\).
Here, the base 'b' is not mentioned, so we assume it to be 10 (common logarithm).
Hence, \(\text{log}_{10} 7 = 0.845\) converts to \(\text{10}^{0.845} = 7\). This shows that by using the definition (logarithms and exponentials being inverses), the conversion becomes straightforward.
Base of logarithms
The base of a logarithm is a crucial component in both logarithmic and exponential equations. Typically, bases are either common logarithms (base 10) or natural logarithms (base e 鈮� 2.718). When the base is not specified, as in \(\text{log} 7 = 0.845\), it is generally assumed to be 10.
The correct interpretation of the base ensures the accuracy of the conversion and subsequent calculations.
Exponential form
The exponential form is another way of expressing a logarithm. Converting a logarithmic equation to its exponential form can often simplify calculations and help solve equations. In our exercise, converting \(\text{log} 7 = 0.845\) to its exponential form involves recognizing that the base (assumed to be 10) raised to the power of the given value equals the number.
Thus, we rewrite it as \(\text{10}^{0.845} = 7\). This exponential form provides a clear and straightforward way to understand the relationship between the base, exponent, and result.

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Most popular questions from this chapter

Use a graphing calculator to find the approximate solutions of the equation. $$x e^{3 x}-1=3$$

Solve using any method. Given that \(a=\log _{8} 225\) and \(b=\log _{2} 15,\) express as a function of \(b\).

Salvage Value. \(\quad\) A landscape company purchased a backhoe for \(\$ 56,395 .\) The value of the backhoe each year is \(90 \%\) of the value of the preceding year. After t years, its value, in dollars, is given by the exponential function $$ V(t)=56,395(0.9)^{t} $$ a) Graph the function. b) Find the value of the backhoe after \(0,1,3,6,\) and 10 years. Round to the nearest dollar.

The following formula can be used to convert Fahrenheit temperatures \(x\) to Celsius temperatures \(T(x):\) $$ T(x)=\frac{5}{9}(x-32) $$ a) Find \(T\left(-13^{\circ}\right)\) and \(T\left(86^{\circ}\right)\) b) Find \(T^{-1}(x)\) and explain what it represents.

Centenarian Population. The centenarian population in the United States has grown over \(65 \%\) in the last 30 years. In \(1980,\) there were only \(32,194\) residents ages 100 and over. This number had grown to \(53,364\) by \(2010 .\) (Sources: Population Projections Program; U.S. Census Bureau; U.S. Department of Commerce; "What People Who Live to 100 Have in Common," by Emily Brandon, U.S. News and World Report, January \(7,2013\) ) The exponential function $$ H(t)=80,040.68(1.0481)^{t} $$ where \(t\) is the number of years after \(2015,\) can be used to project the number of centenarians. Use this function to project the centenarian population in 2020 and in 2050 (IMAGE CANT COPY)
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