5^(logx) + x^(log5) = 50 . Find x
5^(logx) + x^(log5) = 50
Note that
log(5^logx)) = logx * log5
log(x^log5)) = log5 * logx
So, the two are equal. That means
2*5^logx = 50
5^logx = 25 = 5^2
logx = 2
x = b^2
where logs are base b.
So, if natural logs, x = e^2
if common logs, x = 10^2 = 100
check (common logs):
5^log100 + 100^log5 = 50
5^2 + 10^log25 = 50
25 + 25 = 50
To find the value of x in the equation 5^(logx) + x^(log5) = 50, we can use logarithmic properties to simplify the equation. Here's the step-by-step process:
Step 1: Let's start by taking the logarithm of both sides of the equation. We'll use the natural logarithm (ln) for this example, but you can use any logarithm base you prefer.
ln(5^(logx) + x^(log5)) = ln(50)
Step 2: Now, we can apply the exponent rule of logarithms, which states that log(a^b) = b * log(a). Using this rule, we can simplify the left side of the equation:
(logx) * ln(5) + (log5) * ln(x) = ln(50)
Step 3: Next, let's use the fact that log(a^b) = b * log(a) is equivalent to log-base-a(a^b) = b. Applying this rule, we can rewrite the equation as:
ln(x) * log-base-5(5) + log-base-5(x) = ln(50)
Step 4: We know that log-base-a(a) = 1, so log-base-5(5) = 1. Simplifying further:
ln(x) + log-base-5(x) = ln(50)
Step 5: Now, we can use the property that log-base-a(a^b) = b to combine the terms on the left side of the equation:
log-base-5(x * x) = ln(50)
Step 6: Simplifying the equation further:
log-base-5(x^2) = ln(50)
Step 7: To remove the logarithm, we can rewrite the equation using exponential form:
x^2 = 5^(ln(50))
Step 8: Evaluate the right side of the equation using a calculator:
x^2 ≈ 50.68
Step 9: Finally, take the square root of both sides to solve for x:
x ≈ ±√50.68
Therefore, the approximate values of x that satisfy the equation 5^(logx) + x^(log5) = 50 are x ≈ +√50.68 and x ≈ -√50.68.