Final answer:
To solve the equation log₅ (4x-7)=4, we convert it to the exponential form 5⁴ = 4x - 7, simplify the equation, and isolate x to find x = 158.
Step-by-step explanation:
To find the exact solution algebraically for the equation log₅ (4x-7)=4, we need to apply the properties of logarithms to solve for the variable x.
First, we convert the logarithmic equation to its exponential form. This is based on the definition of a logarithm, which in this case can be written as 5 to the power of 4 equals 4x - 7:
5⁴ = 4x - 7
Now, simplify the equation by performing the exponentiation:
625 = 4x - 7
Next, we apply basic algebra to isolate x:
625 + 7 = 4x632 = 4x
Divide both sides of the equation by 4 to solve for x:
x = 158
We have found the solution, x = 158.