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Guias de estudo > College Algebra

Convert Between Logarithmic And Exponential Form

Learning Objectives

  • Convert a logarithm to exponential form
  • Convert an exponential expression to logarithmic form
In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is 10x=500{10}^{x}=500, where x represents the difference in magnitudes on the Richter Scale. How would we solve for x? We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve 10x=500{10}^{x}=500. We know that 102=100{10}^{2}=100 and 103=1000{10}^{3}=1000, so it is clear that x must be some value between 2 and 3, since y=10xy={10}^{x} is increasing. We can examine a graph to better estimate the solution. Graph of the intersections of the equations y=10^x and y=500. Estimating from a graph, however, 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 y=bxy={b}^{x} is one-to-one, so its inverse, x=byx={b}^{y} 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 function. To represent y as a function of x, we use a logarithmic function of the form y=logb(x)y={\mathrm{log}}_{b}\left(x\right). The base b logarithm of a number is the exponent by which we must raise b to get that number. We read a logarithmic expression as, "The logarithm with base b of x 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 example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since 25=32{2}^{5}=32, we can write log232=5{\mathrm{log}}_{2}32=5. We read this as "log base 2 of 32 is 5." We can express the relationship between logarithmic form and its corresponding exponential form as follows: logb(x)=y⇔by=x,b>0,b≠1{\mathrm{log}}_{b}\left(x\right)=y\Leftrightarrow {b}^{y}=x,\text{}b>0,b\ne 1 Note that the base b is always positive. Think b to the y equals x. Because logarithm is a function, it is most correctly written as logb(x){\mathrm{log}}_{b}\left(x\right), using parentheses to denote function evaluation, just as we would with f(x)f\left(x\right). 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 logbx{\mathrm{log}}_{b}x. Note that many calculators require parentheses around the x. We can illustrate the notation of logarithms as follows: logb (c) = a means b to the A power equals C. Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means y=logb(x)y={\mathrm{log}}_{b}\left(x\right) and y=bxy={b}^{x} are inverse functions.

A General Note: Definition of the Logarithmic Function

A logarithm base b of a positive number x satisfies the following definition. For x>0,b>0,b≠1x>0,b>0,b\ne 1, y=logb(x) is equivalent to by=xy={\mathrm{log}}_{b}\left(x\right)\text{ is equivalent to }{b}^{y}=x where,
  • we read logb(x){\mathrm{log}}_{b}\left(x\right) as, "the logarithm with base b of x" or the "log base b of x."
  • the logarithm y is the exponent to which b must be raised to get x.
Also, 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 function. Therefore,
  • the domain of the logarithm function with base b is(0,∞)b \text{ is} \left(0,\infty \right).
  • the range of the logarithm function with base b is(−∞,∞)b \text{ is} \left(-\infty ,\infty \right).

Q & A

Can we take the logarithm of a negative number?

No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.

How To: Given an equation in logarithmic form logb(x)=y{\mathrm{log}}_{b}\left(x\right)=y, convert it to exponential form.

  1. Examine the equation y=logbxy={\mathrm{log}}_{b}x and identify b, y, and x.
  2. Rewrite logbx=y{\mathrm{log}}_{b}x=y as by=x{b}^{y}=x.

Example: Converting from Logarithmic Form to Exponential Form

Write the following logarithmic equations in exponential form.
  1. log6(6)=12{\mathrm{log}}_{6}\left(\sqrt{6}\right)=\frac{1}{2}
  2. log3(9)=2{\mathrm{log}}_{3}\left(9\right)=2

Answer: First, identify the values of b, y, and x. Then, write the equation in the form by=x{b}^{y}=x.

  1. log6(6)=12{\mathrm{log}}_{6}\left(\sqrt{6}\right)=\frac{1}{2} Here, b=6,y=12,and x=6b=6,y=\frac{1}{2},\text{and } x=\sqrt{6}. Therefore, the equation log6(6)=12{\mathrm{log}}_{6}\left(\sqrt{6}\right)=\frac{1}{2} is equivalent to 612=6{6}^{\frac{1}{2}}=\sqrt{6}.
  2. log3(9)=2{\mathrm{log}}_{3}\left(9\right)=2 Here, b = 3, y = 2, and x = 9. Therefore, the equation log3(9)=2{\mathrm{log}}_{3}\left(9\right)=2 is equivalent to 32=9{3}^{2}=9.

Try It

Write the following logarithmic equations in exponential form.
  1. log10(1,000,000)=6{\mathrm{log}}_{10}\left(1,000,000\right)=6
  2. log5(25)=2{\mathrm{log}}_{5}\left(25\right)=2

Answer:

  1. log10(1,000,000)=6{\mathrm{log}}_{10}\left(1,000,000\right)=6 is equivalent to 106=1,000,000{10}^{6}=1,000,000
  2. log5(25)=2{\mathrm{log}}_{5}\left(25\right)=2 is equivalent to 52=25{5}^{2}=25

Convert from exponential to logarithmic form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write x=logb(y)x={\mathrm{log}}_{b}\left(y\right).

Example: Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.
  1. 23=8{2}^{3}=8
  2. 52=25{5}^{2}=25
  3. 10−4=110,000{10}^{-4}=\frac{1}{10,000}

Answer: First, identify the values of b, y, and x. Then, write the equation in the form x=logb(y)x={\mathrm{log}}_{b}\left(y\right).

  1. 23=8{2}^{3}=8 Here, b = 2, x = 3, and y = 8. Therefore, the equation 23=8{2}^{3}=8 is equivalent to log2(8)=3{\mathrm{log}}_{2}\left(8\right)=3.
  2. 52=25{5}^{2}=25 Here, b = 5, x = 2, and y = 25. Therefore, the equation 52=25{5}^{2}=25 is equivalent to log5(25)=2{\mathrm{log}}_{5}\left(25\right)=2.
  3. 10−4=110,000{10}^{-4}=\frac{1}{10,000} Here, b = 10, x = –4, and y=110,000y=\frac{1}{10,000}. Therefore, the equation 10−4=110,000{10}^{-4}=\frac{1}{10,000} is equivalent to log10(110,000)=−4{\text{log}}_{10}\left(\frac{1}{10,000}\right)=-4.

Try It

Write the following exponential equations in logarithmic form.
  1. 32=9{3}^{2}=9
  2. 53=125{5}^{3}=125
  3. 2−1=12{2}^{-1}=\frac{1}{2}

Answer:

  1. 32=9{3}^{2}=9 is equivalent to log3(9)=2{\mathrm{log}}_{3}\left(9\right)=2
  2. 53=125{5}^{3}=125 is equivalent to log5(125)=3{\mathrm{log}}_{5}\left(125\right)=3
  3. 2−1=12{2}^{-1}=\frac{1}{2} is equivalent to log2(12)=−1{\text{log}}_{2}\left(\frac{1}{2}\right)=-1

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