For log(5)x^8 = 16, are both 25 and -25 solutions?
To determine if both 25 and -25 are solutions to the equation log(5)x^8 = 16, we need to substitute each value into the equation and check if the equation holds true.
Let's start with the first value, x = 25:
Substituting x = 25 into the equation, we get:
log(5)(25^8) = 16
Simplifying further:
log(5)(390625) = 16
Now, to solve for log(5)(390625), we need to rewrite the equation in exponential form. In logarithmic form, log(a)b = c, the equivalent exponential form is b = a^c.
So, in this case, we have:
390625 = 5^16
Checking if this equation holds true:
390625 = 390625
Since the equation holds true for x = 25, it is a solution to the original equation.
Now, let's substitute x = -25:
Substituting x = -25 into the equation, we get:
log(5)(-25^8) = 16
Here, we run into a problem. Logarithms are only defined for positive numbers, so taking the log of a negative number doesn't make sense.
Therefore, x = -25 is not a valid solution to the original equation.
In conclusion, only x = 25 is a solution to the equation log(5)x^8 = 16.