Solve the logarithmic equation.
[tex]\log _6 x+\log _6 21=1[/tex]
x =
(Simplify your answer. Type an exact answer, using $e$ as needed.)
Answer (2)
Use the logarithm property lo g b m + lo g b n = lo g b mn to rewrite the equation as lo g 6 ( 21 x ) = 1 .
Convert the logarithmic equation to exponential form: 6 1 = 21 x .
Solve for x by dividing both sides by 21: x = 21 6 .
Simplify the fraction to get the final answer: 7 2 .
Explanation
Problem Analysis We are given the logarithmic equation lo g 6 x + lo g 6 21 = 1 . Our goal is to solve for x .
Using Logarithm Properties We can use the logarithm property that states the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments. That is, lo g b m + lo g b n = lo g b mn . Applying this property to our equation, we get: lo g 6 x + lo g 6 21 = lo g 6 ( x ⋅ 21 ) = lo g 6 ( 21 x ) So our equation becomes lo g 6 ( 21 x ) = 1 .
Converting to Exponential Form Now we need to convert the logarithmic equation to an exponential equation. Recall that lo g b a = c is equivalent to b c = a . Applying this to our equation lo g 6 ( 21 x ) = 1 , we get: 6 1 = 21 x So, 6 = 21 x .
Solving for x To solve for x , we divide both sides of the equation by 21: x = 21 6 Now we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 6 and 21 is 3. Dividing both the numerator and the denominator by 3, we get: x = 21 ÷ 3 6 ÷ 3 = 7 2
Final Answer Therefore, the solution to the equation lo g 6 x + lo g 6 21 = 1 is x = 7 2 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, measuring sound intensity in decibels, and determining the pH of a solution in chemistry. For example, if you know the intensity of an earthquake and want to find its magnitude, you would use a logarithmic equation. Similarly, in finance, logarithmic equations can be used to model the growth of investments over time.
The solution to the equation lo g 6 x + lo g 6 21 = 1 is x = 7 2 . This is found by using logarithm properties to combine the logs, converting to exponential form, and solving for x . Finally, we minimize our fraction to achieve the simplest form.
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