Just learned log rules and I don't get it. How do you solve log5x^2 + 4log5x = 12 ? Thanks in advance.
log_5(x^2) + 4log_5(x) = 12
2log_5(x) + 4log_5(x) = 12
6log_5(x) = 12
log_5(x) = 2
x = 5^2 = 25
To solve the equation log5(x^2) + 4log5(x) = 12, we can use the properties of logarithms to simplify it. Here are the steps:
Step 1: Apply the power rule of logarithms.
2log5(x) + 4log5(x) = 12
Step 2: Combine the logarithms using the product rule.
6log5(x) = 12
Step 3: Divide both sides of the equation by 6 to isolate the logarithm.
log5(x) = 2
Step 4: Rewrite the equation in exponential form.
x = 5^2
Step 5: Simplify the exponential expression.
x = 25
Therefore, the solution to the equation log5(x^2) + 4log5(x) = 12 is x = 25.
To solve the equation log5(x^2) + 4log5(x) = 12, we can use the properties of logarithms. Specifically, the logarithmic rule log(a^b) = b * log(a) and the rule log(ab) = log(a) + log(b).
Let's break down the problem step by step:
1. Start by applying the log(a^b) rule to log5(x^2):
log5(x^2) = 2 * log5(x)
2. Now, substitute this into the original equation:
2 * log5(x) + 4log5(x) = 12
3. Combine the terms that have the same base, which in this case is log5(x):
2 * log5(x) + 4 * log5(x) = 12
(2 + 4) * log5(x) = 12
6 * log5(x) = 12
4. Divide both sides by 6 to isolate log5(x):
log5(x) = 12/6
log5(x) = 2
Now, to get rid of the logarithm on the left side of the equation, we can rewrite it using exponential notation, which means x = 5^2:
5. Take the exponential of both sides:
x = 5^2
x = 25
So, the solution to the equation log5(x^2) + 4log5(x) = 12 is x = 25.