(1/5)^(x-2)=125
If you think about it, 1/5 is really 5^-1, which is read "five raised to the negative one."
125 becomes 5^3.
We needed to have the SAME BASE of 5 on BOTH sides of the exponential equation.
We now have this:
5^-1(x - 2) = 5^3
NEXT:
Remove the parentheses from the left side by applying the distributive rule on the exponent.
-1(x - 2) = -x + 2.
We now have this:
5^(-x + 2) = 5^3
Here comes the easy part--->bring down the exponents and solve for x.
-x + 2 = 3
-x = 3 - 2
-x = 1
x = 1/-1
x = -1
How do we know that x = -1?
Let's plug -1 for x in the ORIGINAL question given and simplify.
If we get the same answer of 125 on BOTH sides, then we will know
that x = -1.
Ready?
You were given this:
1/5^(x - 2) = 125
Let x = -1.
1/5(-1 - 2) = 125
1/5^(-3) = 125
1 divided by 1/5^3 = 125
1 divided by 1/125 = 125
125/1 = 125
125 = 125
It checks!!!
Final answer:
x = -1
Done!
To solve the equation (1/5)^(x-2) = 125, we need to eliminate the exponent by using logarithms. In this case, we'll use the logarithm base 1/5 because it matches the base of the exponential equation.
First, let's rewrite the equation using logarithmic notation:
log[1/5] 125 = (x - 2)
Now we can solve for x by isolating it:
x - 2 = log[1/5] 125
To evaluate the logarithm, we can convert it to a different base by using the logarithmic change of base formula. For example, we can use the natural logarithm (ln) or common logarithm (log base 10) since those are commonly available on calculators.
Let's use the logarithmic change of base formula to convert the logarithm to base 10:
x - 2 = log[10] 125 / log[10] (1/5)
Now, we can simplify further:
x - 2 = log[10] 125 / log[10] 1 - log[10] 5
Since log[10] 1 is equal to 0, we can simplify the equation:
x - 2 = -log[10] 5
Finally, to isolate x, we add 2 to both sides:
x = 2 - log[10] 5
Now we have the solution for x in terms of a logarithm. You can use a calculator to compute the value of log[10] 5 and then subtract it from 2 to find the numerical value of x.