91- 6303239042
$$\log_{a}{1} = 0$$
$$\log_{a}{xy} = \log_{a}{x}+\log_{a}{y}$$
$$\log_{a}{b}^{c} = c \log_{a}{b}$$
$$\log_{b^n}{a}=\dfrac{1}{n}\log_ba\ $$
$$log_a(x/y) = log_a(x) − log_a(y)$$
$$log_a(a) = 1$$
Sign in
Please select an account to continue using cracku.in
↓ →
$$\log_{a}{1} = 0$$
$$\log_{a}{xy} = \log_{a}{x}+\log_{a}{y}$$
$$\log_{a}{b}^{c} = c \log_{a}{b}$$
$$\log_{b^n}{a}=\dfrac{1}{n}\log_ba\ $$
$$log_a(x/y) = log_a(x) − log_a(y)$$
$$log_a(a) = 1$$
Formula Video
If $$5 - \log_{10}\sqrt{1 + x} + 4 \log_{10} \sqrt{1 - x} = \log_{10} \frac{1}{\sqrt{1 - x^2}}$$, then 100x equals
Correct Answer: 99
$$5 - \log_{10}\sqrt{1 + x} + 4 \log_{10} \sqrt{1 - x} = \log_{10} \frac{1}{\sqrt{1 - x^2}}$$
We can re-write the equation as: $$5-\log_{10}\sqrt{1+x}+4\log_{10}\sqrt{1-x}=\log_{10}\left(\sqrt{1+x}\times\ \sqrt{1-x}\right)^{-1}$$
$$5-\log_{10}\sqrt{1+x}+4\log_{10}\sqrt{1-x}=\left(-1\right)\log_{10}\left(\sqrt{1+x}\right)+\left(-1\right)\log_{10}\left(\sqrt{1-x}\right)$$
$$5=-\log_{10}\sqrt{1+x}+\log_{10}\sqrt{1+x}-\log_{10}\sqrt{1-x}-4\log_{10}\sqrt{1-x}$$
$$5=-5\log_{10}\sqrt{1-x}$$
$$\sqrt{1-x}=\frac{1}{10}$$
Squaring both sides: $$\left(\sqrt{1-x}\right)^2=\frac{1}{100}$$
$$\therefore\ $$ $$x=1-\frac{1}{100}=\frac{99}{100}$$
Hence, $$100\ x\ =100\times\ \frac{99}{100}=99$$
For all possible integers n satisfying $$2.25\leq2+2^{n+2}\leq202$$, then the number of integer values of $$3+3^{n+1}$$ is:
Correct Answer: 7
$$2.25\leq2+2^{n+2}\leq202$$
$$2.25-2\le2+2^{n+2}-2\le202-2$$
$$0.25\le2^{n+2}\le200$$
$$\log_20.25\le n+2\le\log_2200$$
$$-2\le n+2\le7.xx$$
$$-4\le n\le7.xx-2$$
$$-4\le n\le5.xx$$
Possible integers = -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
If we see the second expression that is provided, i.e
$$3+3^{n+1}$$, it can be implied that n should be at least -1 for this expression to be an integer.
So, n = -1, 0, 1, 2, 3, 4, 5.
Hence, there are a total of 7 values.
For a real number a, if $$\frac{\log_{15}{a}+\log_{32}{a}}{(\log_{15}{a})(\log_{32}{a})}=4$$ then a must lie in the range
We have :$$\frac{\log_{15}{a}+\log_{32}{a}}{(\log_{15}{a})(\log_{32}{a})}=4$$
We get $$\frac{\left(\frac{\log a}{\log\ 15}+\frac{\log a}{\log32}\right)}{\frac{\log a}{\log\ 15}\times\ \frac{\log a}{\log32}\ \ }=4$$
we get $$\log a\left(\log32\ +\log\ 15\right)=4\left(\log\ a\right)^2$$
we get $$\left(\log32\ +\log\ 15\right)=4\log a$$
=$$\log480=\log a^4$$
=$$a^4\ =480$$
so we can say a is between 4 and 5 .
If $$x$$ and $$y$$ are positive real numbers such that $$\log_{x}(x^2 + 12) = 4$$ and $$3 \log_{y} x = 1$$, then $$x + y $$ equals
Given, $$\log_{x}(x^2 + 12) = 4$$
=> $$x^2+12=x^4$$
=> $$x^4-x^2-12=0$$
=> $$x^4-4x^2+3x^2-12=0$$
=> $$x^2\left(x^2-4\right)+3\left(x^2-4\right)=0$$
=> $$\left(x^2-4\right)\left(x^2+3\right)=0$$ => since, x is a positive real number (given) => x = 2.
Now, Given $$3 \log_{y} x = 1$$
=> $$\log_yx=\frac{1}{3}$$
=> $$x=y^{\frac{1}{3}}$$
=> $$y=x^3$$ => y = 8.
=> x + y = 2 + 8 = 10.
Log in to view all questions
Login| Name | Date Taken | Score | Status |
|---|---|---|---|
| Formula Test 1 | NA | NA | |
| Formula Test 2 | NA | NA | |
| Formula Test 3 | NA | NA | |
| Formula Test 4 | NA | NA | |
| Formula Test 5 | NA | NA | |
| Formula Test 6 | NA | NA | |
| Formula Test 7 | NA | NA | |
| Formula Test 8 | NA | NA | |
| Formula Test 9 | NA | NA | |
| Formula Test 10 | NA | NA | |
| Formula Test 11 | NA | NA | |
| Formula Test 12 | NA | NA | |
| Formula Test 13 | NA | NA | |
| Formula Test 14 | NA | NA | |
| Formula Test 15 | NA | NA | |
| Formula Test 16 | NA | NA | |
| Formula Test 17 | NA | NA | |
| Formula Test 18 | NA | NA | |
| Formula Test 19 | NA | NA | |
| Formula Test 20 | NA | NA | |
| Formula Test 21 | NA | NA | |
| Formula Test 22 | NA | NA | |
| Formula Test 23 | NA | NA | |
| Formula Test 24 | NA | NA | |
| Formula Test 25 | NA | NA | |
| Formula Test 26 | NA | NA | |
| Formula Test 27 | NA | NA | |
| Formula Test 28 | NA | NA | |
| Formula Test 29 | NA | NA | |
| Formula Test 30 | NA | NA | |
| Formula Test 31 | NA | NA | |
| Formula Test 32 | NA | NA | |
| Formula Test 33 | NA | NA | |
| Formula Test 34 | NA | NA | |
| Formula Test 35 | NA | NA | |
| Formula Test 36 | NA | NA | |
| Formula Test 37 | NA | NA | |
| Formula Test 38 | NA | NA |
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Not attempted
Terms of Service
Start your IIM journey with the right preparation and crack CAT 2026.
Educational materials for CAT preparation
Quick, Easy and Effective Revision
By proceeding you agree to create your account
Free CAT Formulas PDF will be sent to your email address soon !!!