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

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

This is about the 4th time I have seen this question in the last couple of days

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BTW, the way you typed it, they are the same equation.
Both have the same solution.

Log(3x+8)=1+log(x-7)

To solve the equations log3(x - 1) = 4 and log3(x - 1) = log3^4, we will follow the steps below for each equation:

Equation 1: log3(x - 1) = 4

Step 1: Start with the given equation:
log3(x - 1) = 4

Step 2: Use the definition of logarithms to rewrite the equation in exponential form:
3^4 = x - 1

Step 3: Simplify the exponential expression:
81 = x - 1

Step 4: Solve for x by isolating it on one side of the equation:
x = 81 + 1
x = 82

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

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

Step 1: Start with the given equation:
log3(x - 1) = log3^4

Step 2: Apply the property of logarithms that states "log(base a) b = log(base a) c implies b = c" to eliminate the logarithms on both sides:
x - 1 = 4

Step 3: Solve for x by isolating it on one side of the equation:
x = 4 + 1
x = 5

Therefore, the solution to the equation log3(x - 1) = log3^4 is x = 5.

To explain the differences between solving the two equations, let's look at each one step by step:

Equation 1: log3(x - 1) = 4

Step 1: Understanding logarithms:
A logarithm is an operation that determines the exponent to which a given base must be raised to obtain a certain number. In this case, we have a base of 3 and an unknown exponent represented by (x - 1).

Step 2: Converting to exponential form:
To solve a logarithmic equation, we can convert it into exponential form. The exponential form of log3(x - 1) = 4 would be 3^4 = x - 1.

Step 3: Simplifying the equation:
Simplifying, we have 81 = x - 1.

Step 4: Solving for x:
To isolate the variable x, we add 1 to both sides of the equation. This gives us 82 = x, so x = 82.

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

Step 1: Understanding the equation:
In this equation, we have log3(x - 1) on the left side and log3^4 on the right side. In logarithmic form, logb(a) = logb(c) implies a = c. So, we can rewrite this equation as x - 1 = 4.

Step 2: Solving for x:
To isolate x, we can add 1 to both sides, giving us x = 5.

In summary, the main difference between these two equations is that in the first equation, we solve by converting the logarithmic equation into exponential form, while in the second equation, we use the property of logarithms to simplify the equation directly.