6^(2log(x)) = 8
assuming logs base 10,
2logx = log_6(8)
logx = 1/2 log_6(8)
x = √(10^(log_6(8))
now log_6(8) = log6/log8. so
x = √(10^log6)^(1/log8)
= √(6^1/log8)
= 6^(1/log64)
take log of both sides
log [ 6^(2logx) ] = log8
2logx (log6) = log8
2logx = log8/log6
log x^2= log8/log6
x^2= 10^(log8/log6)
x = √( 10^(log8/log6) ) = 3.804339012
check using your calculator, it works
Go with Reiny's answer. I messed up my log base conversion.
To solve the equation 6^(2log(x)) = 8, we can use the properties of logarithms and exponentials. Let's break it down step by step:
Step 1: Rewrite 8 as a power of 6.
Since 6 is the base of the exponent on the left side of the equation, we want to express 8 as a power of 6. We know that 6^2 equals 36, and 6^3 equals 216. Since 8 is between these two values, we can approximate it as 6^1.89.
Step 2: Set up the equation.
After rewriting 8 as a power of 6, we have:
6^(2log(x)) = 6^1.89
Step 3: Apply the logarithm property.
To solve for x, we can use the property that if a = b, then log base a of b is equal to 1. Therefore, we can write:
2log(x) = 1.89
Step 4: Solve for log(x).
Divide both sides of the equation by 2 to isolate log(x):
log(x) = 1.89 / 2
log(x) = 0.945
Step 5: Convert logarithmic form to exponential form.
Rewrite the equation in exponential form:
x = 10^0.945
Step 6: Evaluate the exponential form.
Using a calculator, evaluate 10^0.945, which equals approximately 9.394.
So, the solution to the equation 6^(2log(x)) = 8 is x = 9.394.