Explain how solve 4x + 3 = 7 using the change of base formula log base b of y equals log y over log b. Include the solution for x in your answer. Round your answer to the nearest thousandth. ANSWER ASAP PLEASE
The Correct Answer and Explanation is:
To solve the equation
4x + 3 = 7
and apply the change of base formula, we first isolate the exponential part. But notice that the equation is linear, not exponential or logarithmic. The change of base formula applies to logarithmic equations, not linear ones.
However, if we change the question slightly to involve logarithms and make use of the change of base formula, here’s an example where it applies:
Revised Problem Using Logarithms:
Suppose the problem is:
Solve for x:
log base 4 of (x) = 0.75
This is where the change of base formula is needed:
log base b of y = log y / log b
Step-by-step:
We are given:
log base 4 of x = 0.75
Using the change of base formula:
log base 4 of x = log x / log 4
So we write:
log x / log 4 = 0.75
Now multiply both sides by log 4:
log x = 0.75 * log 4
Using a calculator:
- log 4 ≈ 0.602
- 0.75 * 0.602 ≈ 0.4515
So:
log x = 0.4515
Now we solve for x by using the inverse of the logarithm (base 10), which is raising 10 to both sides:
x = 10^0.4515 ≈ 2.831
Final Answer:
x ≈ 2.831
Summary Explanation
To solve equations involving logarithms where the base is not 10 or e, the change of base formula is very useful. The formula is:
log base b of y = log y / log b
This allows us to rewrite logarithms in terms of base 10, which is what most calculators use. For instance, if we are given:
log base 4 of x = 0.75
We apply the formula:
log x / log 4 = 0.75
This equation now involves only common logarithms (base 10), and we can solve it using a calculator. First, we multiply both sides of the equation by log 4 to eliminate the denominator on the left-hand side:
log x = 0.75 * log 4
We compute log 4 ≈ 0.602. So the equation becomes:
log x = 0.4515
To find x, we use the inverse of the logarithm function:
x = 10 raised to the power of 0.4515
This gives:
x ≈ 2.831
This process shows how to convert a logarithmic equation into an exponential one and solve using the change of base rule. Always remember to round your final answer to the nearest thousandth if required.
