Final answer:
To write ln(1000x^4) as a sum of logarithms, it can be expressed as 3 × ln(10) + 4 × ln(x), using the product and power rules for logarithms.
Step-by-step explanation:
When asked to write ln(1000x4) as a sum, difference or multiple of logarithms, we can apply several logarithmic properties. First, recall the property that the logarithm of a product can be written as the sum of the logarithms: ln(xy) = ln(x) + ln(y). Then, use the property that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base: ln(xn) = n × ln(x). Applying these rules to ln(1000x4), we get:
- ln(1000) + ln(x4) because of the product rule for logarithms.
- ln(1000) + 4 × ln(x) because of the power rule for logarithms.
Finally, since 1000 is a power of 10 (103), we can write ln(1000) as 3 × ln(10), assuming the student is familiar with the natural logarithm of 10. So, the original expression ln(1000x4) can be expressed as 3 × ln(10) + 4 × ln(x).