Find x if log$_9$x = 1.5

Analysis

The question asks us to solve for $x$ in the logarithmic equation $\log_9 x = 1.5$. To solve this, we need to convert the logarithmic form into its equivalent exponential form.

Concept: Logarithms to Exponents

The fundamental definition of a logarithm states that if $\log_b a = c$, then $b^c = a$. In this equation:

• The base ($b$) is 9
• The exponent ($c$) is 1.5
• The result ($a$) is $x$

Step-by-Step Solution

Step 1: Rewrite in exponential form Using the definition above: $x = 9^{1.5}$

Step 2: Simplify the exponent Recall that $1.5$ can be written as a fraction: $1.5 = \frac{3}{2}$. So, $x = 9^{3/2}$

Step 3: Evaluate the expression The fractional exponent $\frac{3}{2}$ means we take the square root of the base and then raise it to the power of 3: $x = (\sqrt{9})^3$

Since $\sqrt{9} = 3$: $x = 3^3$

Step 4: Calculate the final value $x = 3 \times 3 \times 3$ $x = 27$

Conclusion

Comparing our result to the given options, we find that $x = 27$ corresponds to option A.

Exam Tip

When you see a decimal exponent like $1.5$ or $0.5$, always convert it to a fraction ($3/2$ or $1/2$) immediately. It makes the calculation much easier to visualize! For example, $x^{0.5}$ is just the square root of $x$.