How To Change From Log To Exponential

Learning Outcomes

• Catechumen from logarithmic to exponential form.
• Catechumen from exponential to logarithmic grade.

In order to analyze the magnitude of earthquakes or compare the magnitudes of two dissimilar earthquakes, we need to be able to convert betwixt logarithmic and exponential form. For example, suppose the amount of energy released from i earthquake was 500 times greater than the amount of energy released from some other. We desire to calculate the divergence in magnitude. The equation that represents this problem is [latex]{ten}^{x}=500[/latex] where 10 represents the departure in magnitudes on the Richter Scale. How would nosotros solve forx?

Nosotros have not yet learned a method for solving exponential equations algebraically. None of the algebraic tools discussed so far is sufficient to solve [latex]{10}^{10}=500[/latex]. We know that [latex]{10}^{two}=100[/latex] and [latex]{x}^{iii}=1000[/latex], so it is articulate that ten must be some value between 2 and 3 since [latex]y={10}^{10}[/latex] is increasing. We can examine a graph to improve estimate the solution.

Estimating from a graph, notwithstanding, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph above passes the horizontal line test. The exponential function [latex]y={b}^{x}[/latex] is one-to-ane, and so its inverse, [latex]10={b}^{y}[/latex] is also a function. As is the case with all inverse functions, we simply interchange x and y and solve for y to find the inverse office. To represent y as a function of ten, nosotros use a logarithmic function of the form [latex]y={\mathrm{log}}_{b}\left(ten\right)[/latex]. The base blogarithm of a number is the exponent by which we must heighten b to become that number.

Nosotros read a logarithmic expression every bit, "The logarithm with base b of ten is equal to y," or, simplified, "log base b of x is y." We can also say, "b raised to the power of y is x," because logs are exponents. For case, the base ii logarithm of 32 is v, considering 5 is the exponent we must use to ii to get 32. Since [latex]{ii}^{5}=32[/latex], we can write [latex]{\mathrm{log}}_{2}32=5[/latex]. Nosotros read this equally "log base of operations 2 of 32 is 5."

We tin can express the relationship between logarithmic class and its corresponding exponential grade as follows:

[latex]{\mathrm{log}}_{b}\left(10\right)=y\Leftrightarrow {b}^{y}=x,\text{}b>0,b\ne ane[/latex]

Note that the base of operations b is always positive.

Because a logarithm is a function, information technology is nearly correctly written as [latex]{\mathrm{log}}_{b}\left(x\right)[/latex] using parentheses to denote function evaluation but as nosotros would with [latex]f\left(x\right)[/latex]. However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses as [latex]{\mathrm{log}}_{b}ten[/latex]. Note that many calculators require parentheses around the x.

We tin illustrate the notation of logarithms as follows:

Notice that when comparing the logarithm office and the exponential function, the input and the output are switched. This means [latex]y={\mathrm{log}}_{b}\left(10\right)[/latex] and [latex]y={b}^{x}[/latex] are changed functions.

A General Notation: Definition of the Logarithmic Function

A logarithm base b of a positive number x satisfies the post-obit definition:

For [latex]x>0,b>0,b\ne 1[/latex],

[latex]y={\mathrm{log}}_{b}\left(x\correct)\text{ is equal to }{b}^{y}=x[/latex], where

• we read [latex]{\mathrm{log}}_{b}\left(x\right)[/latex] as, "the logarithm with base b of x" or the "log base b of 10."
• the logarithm y is the exponent to which b must be raised to get x.
• if no base [latex]b[/latex] is indicated, the base of the logarithm is assumed to exist [latex]ten[/latex].

Besides, since the logarithmic and exponential functions switch the x and y values, the domain and range of the exponential function are interchanged for the logarithmic part. Therefore,

• the domain of the logarithm office with base [latex]b \text{ is} \left(0,\infty \right)[/latex].
• the range of the logarithm role with base [latex]b \text{ is} \left(-\infty ,\infty \correct)[/latex].

Q & A

Can nosotros take the logarithm of a negative number?

No. Because the base of an exponential function is ever positive, no power of that base can e'er be negative. Nosotros can never have the logarithm of a negative number. Too, we cannot take the logarithm of zippo. Calculators may output a log of a negative number when in circuitous mode, but the log of a negative number is non a real number.

How To: Given an equation in logarithmic form [latex]{\mathrm{log}}_{b}\left(10\correct)=y[/latex], convert it to exponential class

1. Examine the equation [latex]y={\mathrm{log}}_{b}x[/latex] and identify b, y, and x.
2. Rewrite [latex]{\mathrm{log}}_{b}x=y[/latex] as [latex]{b}^{y}=10[/latex].

Example: Converting from Logarithmic Form to Exponential Course

Write the post-obit logarithmic equations in exponential course.

1. [latex]{\mathrm{log}}_{half-dozen}\left(\sqrt{6}\correct)=\frac{1}{ii}[/latex]
2. [latex]{\mathrm{log}}_{3}\left(ix\right)=two[/latex]

Try It

Write the following logarithmic equations in exponential form.

1. [latex]{\mathrm{log}}_{10}\left(1,000,000\correct)=6[/latex]
2. [latex]{\mathrm{log}}_{5}\left(25\right)=two[/latex]

Convert from Exponential to Logarithmic Class

To convert from exponential to logarithmic form, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write [latex]x={\mathrm{log}}_{b}\left(y\right)[/latex].

Case: Converting from Exponential Form to Logarithmic Course

Write the following exponential equations in logarithmic form.

1. [latex]{2}^{three}=8[/latex]
2. [latex]{5}^{ii}=25[/latex]
3. [latex]{10}^{-4}=\frac{i}{10,000}[/latex]

Effort Information technology

Write the following exponential equations in logarithmic class.

1. [latex]{three}^{two}=9[/latex]
2. [latex]{5}^{3}=125[/latex]
3. [latex]{2}^{-one}=\frac{1}{2}[/latex]

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