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.