If $ b > 0 $ and $ 4\log_x b + 9\log_{b^5x} b = 1 $, then the possible value(s) of $ x $ is (are)
(a) $ b^{10} $ ✓
(b) $ b^9 $
(c) $ b^{-2} $ ✓
(d) $ b^5 $
(e) $ b^4 $
Solution:
Let $ p = \log_b x $.
Equation: $ \frac{4}{p} + \frac{9}{5 + p} = 1 $
Solve: $ p^2 - 8p - 20 = 0 $, roots $ p = -2, 10 $.
So, $ x = b^{-2} $ or $ x = b^{10} $.
Question 2
George deposits ₹5L in a bank that compounds quarterly at 20% per year. How long will it take to increase his money to 16 times the principal (in years)?
(b) If $ 0 < b < 1 $ and $ 0 < x < 1 $, then $ \log_b x < 0 $.
(c) If $ \log_3(\log_5 x) = 1 $, then $ x = 125 $ ✓
(d) If $ 0 < b < 1 $, $ 0 < x < 1 $ and $ x > b $, then $ \log_b x > 1 $.
(e) If $ 0 < b < 1 $ and $ 0 < x < y $, then $ \log_b x > \log_b y $ ✓
Solution:
(c): $ \log_3(\log_5 x) = 1 \implies x = 125 $.
(e): $ \log_b x > \log_b y $ for $ 0 < b < 1 $ and $ 0 < x < y $.
Question 4
Two types of insects grow as $ f(t) = 5^{3t-2} $ and $ h(t) = 3^{2t-1} $. For what value of $ t $ will both insects be of the same number?
(a) $ \frac{\ln 3 + 2\ln 5}{3\ln 5 - 2\ln 3} $
(b) $ \frac{\ln 75}{\ln (125/9)} $
(c) $ \log_{125} 75 $ ✓
Solution:
Set $ 5^{3t-2} = 3^{2t-1} $, solve for $ t $:
$ t = \frac{\ln 3 + 2\ln 5}{3\ln 5 - 2\ln 3} = \log_{125} 75 $.
Question 5
Which of the following is/are true? (MSQ)
(a) If $ m $ and $ n $ are positive real numbers, then $ m^{\log n} = n^{\log m} $ ✓
(b) $ \log_5 1234567899999999999999 $ is a rational number.
(c) The function $ f(x) = \log_{10}(x^2 + x + 1) $ is one-one on $(-0.5, \infty)$ ✓
Solution:
(a): True by taking logs.
(c): $ x^2 + x + 1 $ is one-one (strictly increasing) on $(-0.5, \infty)$, so is its log.
Question 6
Which of the following is/are true? (MSQ)
(a) If $ f $ is one-one on $ D $, then $ \log f(x) $ is one-one on $ D $ where defined ✓
(b) $ (14!) < (15!) $ ✓
(c) $ f(x) = 2^x + 3^x + \dots + 100^x $ is one-one on $ \mathbb{R} $ ✓
(d) There exists a function $ f(x) $ on $ \mathbb{R} $ such that $ \log(f(x)) \geq 100 $ for all $ x \in \mathbb{R} $ ✓
Solution:
All options (a), (b), (c), (d) are correct.
Question 7
If $ \log_2(x + 4) - \log_2\left(\frac{x}{2} + 2\right) = 1 $, then $ x $ is
(a) $-3$ ✓
(b) 1
(c) $-4$
(d) 5
Solution:
Solve: $ x = -3 $ (only valid solution in domain).
Question 8
Seismologists use the Richter scale: $ R = \ln I - \ln I_0 $. If an earthquake in city A is magnitude 8.0 and city B is reference, what is the ratio of intensities?
(a) $ e^8 : 1 $ ✓
(b) $ e^1 : 2 $
(c) $ e^8 : 1 $
(d) $ e^5 : 1 $
(e) $ e^8 : 2 $
Solution:
$ \frac{I}{I_0} = e^8 $, so ratio is $ e^8 : 1 $.
Question 9
Bacteria grow as $ G(t) = G_0 3^{kt} $. If initial count is 1000, after 1 min it is 9000, how long to reach 730,000?
