Explain the differences between solving these two equations:

•log3(x - 1) = 4
AND log3(x - 1) = log34

I think it has to deal with the formulas?

by definition

log3(x - 1) = 4
is 3^4 = x-1
x= 82

for the second, (I will assume you meant
log3(x - 1) = log34 )

anti-log it
x-1 = 4
x = 5

To understand the differences between solving these two equations, let's break it down step by step.

1. log3(x - 1) = 4:
In this equation, the logarithm base is 3. We are trying to find the value of x that makes the logarithm of (x - 1) with base 3 equal to 4.

To solve this equation, we need to use the properties of logarithms. By applying the exponential form of a logarithm, we can rewrite the equation as follows:

3^4 = x - 1

Simplifying further, we have:

81 = x - 1

Now, we solve for x by adding 1 to both sides:

x = 82

Therefore, x = 82 is the solution to the equation log3(x - 1) = 4.

2. log3(x - 1) = log34:
In this equation, the logarithm base is still 3. However, instead of a numerical value on the right-hand side, we have another logarithmic expression. We need to find the value of x that makes the logarithmic expression on both sides equal.

To solve this equation, we can take advantage of the logarithmic property that states if loga(b) = loga(c), then b = c.

Therefore, we can equate the arguments of the logarithms:

x - 1 = 4

Now, solve for x by adding 1 to both sides:

x = 5

Therefore, x = 5 is the solution to the equation log3(x - 1) = log34.

In summary:

- For the equation log3(x - 1) = 4, the solution is x = 82.
- For the equation log3(x - 1) = log34, the solution is x = 5.

The difference lies in the form of the right-hand side: a numerical value versus another logarithmic expression. The approach to solving these equations varies accordingly.