TheCATExam
Sign in

CAT 2021 Slot 3 — QA Question 1

LogarithmsEasy

For a real number a, if log15a+log32alog15a×log32a = 4, then a must lie in the range

Answer & solution

  • 4 < a < 5

  • B

    2 < a < 3

  • C

    a > 5

  • D

    3 < a < 4

Solution

Easy

The given expression is a sum of two reciprocals divided by their product. Splitting the fraction turns it into a sum of "flipped" logarithms, which collapse neatly using the reciprocal rule 1log⁡ba=log⁡ab\dfrac{1}{\log_b a}=\log_a b and the product rule for logarithms. Then estimate the resulting power of aa.

1

Split the fraction. Write the single fraction as a sum of two terms by dividing each part of the numerator by the product in the denominator.

log⁡15a+log⁡32alog⁡15a⋅log⁡32a=4⇒ 1log⁡32a+1log⁡15a=4(cancel one log in each term)\begin{aligned} &\frac{\log_{15}a+\log_{32}a}{\log_{15}a\cdot\log_{32}a}=4\\ &\Rightarrow\ \frac{1}{\log_{32}a}+\frac{1}{\log_{15}a}=4 \quad\text{(cancel one log in each term)} \end{aligned}
2

Flip each logarithm. Use the change-of-base identity 1log⁡ba=log⁡ab\dfrac{1}{\log_b a}=\log_a b, then combine with the product rule.

log⁡a32+log⁡a15=4(reciprocal rule)⇒ log⁡a(32×15)=4(product rule)⇒ log⁡a480=4\begin{aligned} &\log_a 32+\log_a 15=4 \quad\text{(reciprocal rule)}\\ &\Rightarrow\ \log_a(32\times 15)=4 \quad\text{(product rule)}\\ &\Rightarrow\ \log_a 480=4 \end{aligned}
3

Solve for aa. Rewrite the logarithm in exponential form and bracket 480480 between fourth powers.

\begin{aligned} &a^4=480\\ &4^4=256\quad\text{and}\quad 5^4=625\\ &\Rightarrow\ 256<480<625 \quad\text{(so }4
4
CAT 2021 Slot 3 QA Q1: For a real number a, if l o g 15 a + l o g 32 a l o g 15 a &times; l o g 32 a = 4, then a must lie in the rang — Solution | TheCATExam