Question 8

x = b^5

To solve for x in terms of b when log_b(x) = 5, we can rewrite the logarithmic equation in its exponential form, which gives us x = b^5.

A) 5/b

This option suggests that x is equal to 5 divided by b, which does not align with the exponential relationship derived from the logarithmic equation. Therefore, this option is incorrect.

B) 5b

This option proposes that x equals 5 times b. However, based on the properties of logarithms, the correct transformation leads to x being expressed as a power of b, not a multiplication by b. Thus, this option is incorrect.

C) 5^b

This choice indicates that x is equal to 5 raised to the power of b. This is not a valid representation of the logarithmic equation provided, as it does not maintain the relationship between x and b established by the logarithm. Consequently, this option is incorrect.

D) b^5

This option correctly represents the value of x in terms of b. By using the properties of logarithms, we know that if log_b(x) = 5, then x must be equal to b raised to the 5th power, confirming this option as correct.

E) b^10

This option suggests that x is equal to b raised to the 10th power. However, this does not correspond to the logarithmic equation given, and therefore it is incorrect.

Conclusion

The correct answer, x = b^5, accurately reflects the transformation of the logarithmic equation log_b(x) = 5 into its exponential form. All other options fail to capture the necessary relationship between x and b as dictated by the logarithm, making them invalid.