In Equation 2, the logarithm is not isolated before exponentiation. Instead, 3 is subtracted from the entire expression inside the logarithm. This effectively shifts the curve of the logarithmic function down by 3 units. As a result, when we exponentiate 1.4 - 3, we get a much smaller number compared to Equation 1. So , despite looking similar, the difference in how the logarithms are treated leads to significantly different solutions for the two equations.
Here's how to solve the equations in the image using inverse operations and why they have different solutions despite looking similar:
Equation 1:
Isolate the logarithm:
log_10(x + 3) = 1.4
Use the inverse operation (exponentiation with base 10) to undo the logarithm:
x + 3 = 10^(1.4)
Solve for x:
Subtract 3 from both sides: x = 10^(1.4) - 3
Equation 2:
Isolate the logarithm:
log_10(x) + 3 = 1.4
Use the inverse operation (exponentiation with base 10) to undo the logarithm:
x = 10^(1.4 - 3)
Solve for x:
Simplify the exponent: x = 10^(-1.6)
Why the different solutions?
Both equations involve logarithms, but they differ in how the logarithms are treated. In Equation 1, the logarithm is isolated and then exponentiated. This means we raise 10 to the power of 1.4, which gives us a relatively large number.