Explain the differences between solving these two equations:
Brake down the steps.
log3(x - 1) = 4 and log3(x - 1) = log34 )
1. log3(x - 1) = 4.
x - 1 = 3^4,
x = 3^4 + 1 = 81 + 1 = 82.
2. log3(x - 1) = log34.
log3(x - 1) = 1.5315,
x - 1 = 3^1.5315,
x = 3^1.5315 + 1 = 5.379 + 1 = 6.379.
Each problem required the use and understanding 0f logs. So they were basically the same.
2. log3(x - 1) = log3(4).
If two numbers have equal logs, they are equal:
x-1 = 4,
x = 4 + 1 = 5.
To understand the differences between solving these two equations, let's break down each step for both equations.
1. Equation: log3(x - 1) = 4
Step 1: Isolate the logarithm expression
In this equation, the logarithm expression is on the left side. Start by isolating it.
log3(x - 1) = 4
Step 2: Convert the logarithm form to exponential form
This step helps us get rid of the logarithm. Recall that loga(b) = c can be rewritten as a^c = b.
3^4 = x - 1
Step 3: Simplify the exponential equation
Calculate the value of 3^4, which is 81. Then solve for x.
81 = x - 1
Step 4: Solve for x
Add 1 to both sides to isolate x.
82 = x
So, the solution to the equation log3(x - 1) = 4 is x = 82.
2. Equation: log3(x - 1) = log34
Step 1: Apply the logarithmic identity
When two logarithmic expressions with the same base are set equal to each other, their arguments (the values inside the logarithms) must be equal.
x - 1 = 4
Step 2: Solve for x
Add 1 to both sides to isolate x.
x = 5
Therefore, the solution for the equation log3(x - 1) = log34 is x = 5.
In summary, the first equation required us to convert the logarithmic equation into an exponential equation and solve for x, while the second equation involved applying the logarithmic identity and directly solving for x.