How do you solve this exponential equation using logarithms?

2*5^x=85

Do you start like this?

xlog2*5=log85

I don't know what to do after that.

I think I figured it out.

85/2=42.5

log5=log42.5

log42.5/log5

=2.330 to the third decimal

2^5x = 85

Log2^5x = log85
5xlog2 = log85
Solve for x
x = log85/5log2

To solve the exponential equation 2 * 5^x = 85 using logarithms, you can indeed start by taking the logarithm of both sides of the equation.

Begin by taking the logarithm (with any base of your choice) of both sides of the equation:

log(2 * 5^x) = log(85)

Using the logarithmic identity log(a * b) = log(a) + log(b), we can split up the logarithm on the left side as follows:

log(2) + log(5^x) = log(85)

Since log(5^x) is equivalent to x * log(5), we can rewrite the equation as:

log(2) + x * log(5) = log(85)

Now, isolate the variable term by subtracting log(2) from both sides:

x * log(5) = log(85) - log(2)

Next, simplify the right side using the logarithmic identity log(a) - log(b) = log(a/b), and substitute the values of log(5) and log(2) if necessary:

x * log(5) = log(85/2)

At this point, you have an equation where x is multiplied by log(5). To solve for x, divide both sides of the equation by log(5):

x = (log(85/2)) / log(5)

You can now use a calculator or other computational tools to evaluate the right side and find the numerical value of x.

Note that you may obtain a rounded value for x, depending on the accuracy of the logarithmic and division computations.