Final answer:
The value of x in the logarithmic equation log₃ 15x⁻³=18 is found by converting the log to an exponential form and simplifying to get x = 15. the final answer, rounded to the nearest thousandth, is 15.000.
Step-by-step explanation:
To find the value of x in the equation log₃ 15x⁻³=18, we need to first convert the log equation into an exponential form. The base of the logarithm is 3, so the exponential form of the equation is 318 = 15x⁻³. next we can rewrite the equation to isolate the term with x: 15x⁻³ = 318. then we compare the exponents with base 15 directly.
Let's solve this step-by-step:
- Rewrite the equation in exponential form: 318 = 15x+3.
- Since 15 is 3 times 5, rewrite 15 as 3×5 and apply the power to both: (3×5)x+3 = 3x+3×5x+3.
- Set the powers of 3 equal to each other: 318 = 3x+3, which simplifies to 18 = x+3.
- Solve for x: x = 18 - 3.
- The value of x is 15.
Therefore the value of x rounded to the nearest thousandth as requested, is 15.000.