What is the solution to $\log _5(10 x-1)=\log _5(9 x+7)$?
A. $x=\frac{6}{19}$
B. $x=\frac{8}{19}$
C. $x=7$
D. $x=8$
Answer (2)
Equate the arguments of the logarithms: 10 x − 1 = 9 x + 7 .
Solve the linear equation for x : x = 8 .
Check if the solution is valid by ensuring the arguments of the logarithms are positive.
The solution is x = 8 .
Explanation
Equating the Arguments We are given the equation lo g 5 ( 10 x − 1 ) = lo g 5 ( 9 x + 7 ) . To solve this equation, we need to find the value of x that makes the equation true. Since the logarithms on both sides of the equation have the same base, we can equate their arguments. This means that 10 x − 1 = 9 x + 7 .
Solving for x Now we have a linear equation: 10 x − 1 = 9 x + 7 . To solve for x , we can subtract 9 x from both sides of the equation: 10 x − 9 x − 1 = 9 x − 9 x + 7 which simplifies to x − 1 = 7 Now, we add 1 to both sides of the equation: x − 1 + 1 = 7 + 1 which gives us x = 8
Checking the Solution Now we need to check if our solution is valid. The domain of the logarithm function requires that the argument of the logarithm must be greater than 0. So, we need to check if 0"> 10 x − 1 > 0 and 0"> 9 x + 7 > 0 when x = 8 .For 10 x − 1 , we have 10 ( 8 ) − 1 = 80 − 1 = 79 , which is greater than 0.For 9 x + 7 , we have 9 ( 8 ) + 7 = 72 + 7 = 79 , which is also greater than 0.Since both arguments are positive when x = 8 , our solution is valid.
Final Answer Therefore, the solution to the equation lo g 5 ( 10 x − 1 ) = lo g 5 ( 9 x + 7 ) is x = 8 .
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution in chemistry, and modeling population growth or decay in biology. For example, if we know the initial and final populations of a bacterial colony and the time it took for the population to change, we can use logarithmic equations to find the growth rate.
The solution to the equation lo g 5 ( 10 x − 1 ) = lo g 5 ( 9 x + 7 ) is x = 8 . This solution is valid as both arguments of the logarithms are positive at this value. Therefore, the correct option is D : x = 8 .
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