1. 2log8^x=log8(4x+12)

both log bases are 8. I have no clue what to do.

2. 1/2ln (x) -3 ln y - ln (z-2)

so far for this second one i have

ln sqrt(x)-ln y^3 - ln (z-2)

dont know what to do next

#1.

assuming base 8, just for ease of readability,

2logx = log(4x+12)
log(x^2) = log(4x+12)
so, if the logs are equal, so are the numbers:
x^2 = 4x+12
x^2-4x-12=0
(x-6)(x+2)=0
x=6 only, since log(-2) is not real.

#2.
so far, so good. Recall that adding/subtracting logs means multiplying/dividing numbers. So, you have

ln sqrt(x)-ln y^3 - ln (z-2)
ln(√x/(y^3(z-2))

That's all you can do.

1. Let's start by simplifying the equation 2log8^x = log8(4x+12).

First, we can use the logarithmic property log_b(a^c) = c * log_b(a) to rewrite the left side as 2x * log8(8). Since log8(8) is equal to 1, the equation can now be simplified to:

2x = log8(4x+12)

Now, we can further simplify the equation by using the change of base formula. The change of base formula states that log_b(a) can be expressed as log_c(a) / log_c(b), where c is any positive real number.

In this case, let's change the base of both sides to the natural logarithm (ln), which is commonly used since most scientific calculators have the ln function. Applying the change of base formula, we get:

2x = ln(4x+12) / ln(8)

To isolate x, we can multiply both sides of the equation by ln(8):

2x * ln(8) = ln(4x+12)

Now, we have:

ln(8^2x) = ln(4x+12)

Using the property of logarithms that ln(a) = ln(b) if and only if a = b, we can conclude that:

8^(2x) = 4x + 12

Now, we have a more simplified equation. From here, we can proceed to solve for x using algebraic techniques such as factoring, solving equations graphically, or employing numerical methods like Newton's method.

2. Let's work on simplifying the expression 1/2ln(x) - 3ln(y) - ln(z-2).

Starting with the first term, we can apply the rule of logarithms that states log_a(c) - log_a(b) = log_a(c/b):

1/2ln(x) - 3ln(y) - ln(z-2)

ln(x^(1/2)) - ln(y^3) - ln(z-2)

ln(sqrt(x)) - ln(y^3) - ln(z-2)

Now, we can simplify further. Using the rule that subtraction of logarithms is equivalent to the logarithm of the quotient:

ln(sqrt(x)/(y^3)) - ln(z-2)

Finally, we can apply the rule that states subtracting logarithms is equivalent to the logarithm of the quotient:

ln((sqrt(x)/(y^3))/(z-2))

This is the simplified form of the expression.