Lesson 3 Assignment .Explain the differences between solving these two equations: By steps

log3(x - 1) = 4
AND
log3(x - 1) = log3^4

To solve the equations log3(x - 1) = 4 and log3(x - 1) = log3^4, let's break down the steps for each equation separately.

Equation 1: log3(x - 1) = 4

Step 1: Remove the logarithm.
To remove the logarithm, we need to rewrite the equation in exponential form. In this case, since the base of the logarithm is 3, we can rewrite the equation as 3^4 = x - 1.

Step 2: Simplify.
Evaluating 3^4 gives us 81, so the equation becomes 81 = x - 1.

Step 3: Solve for x.
To solve for x, we need to isolate it. Adding 1 to both sides of the equation, we get 82 = x.

Therefore, the solution to this equation is x = 82.

Equation 2: log3(x - 1) = log3^4

Step 1: Simplify the logarithmic expression.
Since log3^4 represents the logarithm of 4 to the base 3, we can simplify the equation to log3(x - 1) = 4.

Step 2: Follow the same steps as in Equation 1.
We can now follow the same steps as in Equation 1 to solve for x.

Step 3: Solve for x.
Following the same steps, we rewrite the equation in exponential form: 3^4 = x - 1. Simplifying it gives us 81 = x - 1. Adding 1 to both sides of the equation, we obtain 82 = x.

Therefore, the solution to this equation is also x = 82.

In conclusion, despite the slight difference in notation, the two equations log3(x - 1) = 4 and log3(x - 1) = log3^4 have the same solution, which is x = 82.