Log4(5x – 4) log4(5x – 1)=1

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To solve this logarithmic equation, we can use the properties of logarithms.

The equation is:
log4(5x – 4) + log4(5x – 1) = 1

We can combine the two logarithms into a single logarithm by using the product rule of logarithms which states that log(a) + log(b) = log(ab).

log4((5x – 4)(5x – 1)) = 1
log4(25x^2 - 9x - 20) = 1

To eliminate the logarithm, we need to rewrite the equation in exponential form. Remember that loga(b) = c is equivalent to a^c = b.

So, using this in our equation:

4^1 = 25x^2 - 9x - 20
4 = 25x^2 - 9x - 20
25x^2 - 9x - 24 = 0

Now, we need to solve the quadratic equation 25x^2 - 9x - 24 = 0. This equation can be factored as:

(5x + 4)(5x - 6) = 0

Setting each factor to zero:

5x + 4 = 0
5x = -4
x = -4/5

and

5x - 6 = 0
5x = 6
x = 6/5

So, the solutions to the equation log4(5x – 4) + log4(5x – 1) = 1 are x = -4/5 and x = 6/5.