Logarithm flash cards

1. 1.If $\log_a x = \log_a y$ for base $a>0, a\neq 1$, when can we conclude $x=y$? What subtlety must be checked in JEE problems involving logarithmic equations?

   We can conclude $x=y$ only if both $x>0$ and $y>0$ (domain of log). Equating log arguments without checking domain often introduces extraneous roots — always verify solutions satisfy $x>0, y>0, a>0, a\neq1$.

   Hint: Domain check is essential

2. 2.Find the number of digits in $2^{100}$ given $\log_{10}2 = 0.3010$.

   Number of digits $= \lfloor 100\log_{10}2\rfloor + 1 = \lfloor 30.10\rfloor + 1 = 30+1 = 31$ digits.

   Hint: Digits $=\lfloor n\log_{10}N\rfloor+1$

3. 3.Solve for $x$: $\log_2(x-1) + \log_2(x+1) = 3$.

   Combine: $\log_2[(x-1)(x+1)] = 3 \Rightarrow x^2-1=8 \Rightarrow x^2=9 \Rightarrow x=\pm3$. Since domain requires $x-1>0 \Rightarrow x>1$, only $x=3$ is valid.

   Hint: Check domain before accepting roots

4. 4.What is the value of $\log_a a^x$ for any real $x$, and how does this differ from $a^{\log_a x}$?

   $\log_a a^x = x$ (valid for all real $x$, no domain restriction on $x$). But $a^{\log_a x} = x$ requires $x>0$. These are inverse operations, but the second has a restricted domain.

   Hint: Same identity, different domains

5. 5.If $\log_{18}9 = a$, express $\log_{9}24$ in terms of $a$. What is the key technique used (JEE Advanced style)?

   Since $18=2\cdot9$, $\log_{18}18=1=\log_{18}2+\log_{18}9$, so $\log_{18}2=1-a$. Then $\log_9 2=\dfrac{\log_{18}2}{\log_{18}9}=\dfrac{1-a}{a}$ and $\log_9 3=\dfrac12$. Writing $24=2^3\cdot3$: $\log_9 24=3\log_9 2+\log_9 3=\dfrac{3(1-a)}{a}+\dfrac12=\dfrac{6-5a}{2a}$. Key technique: use the complementary relation $\log_{18}2+\log_{18}9=1$, then change of base to rewrite everything in base 9.

   Hint: Use $\log_{18}2+\log_{18}9=1$

6. 6.Find the domain of $f(x)=\log_{(x-1)}(x+2)$.

   Need base $x-1>0,\ x-1\neq1$ and argument $x+2>0$, giving $x>-2$, $x>1$, $x\neq2$. Combining, the domain is $x\in(1,2)\cup(2,\infty)$.

   Hint: Remember a log base must be positive and not equal to 1.

7. 7.Find the domain of $f(x)=\log_2\big(\log_3(\log_4 x)\big)$.

   We need $\log_4 x>0\Rightarrow x>1$, then $\log_3(\log_4x)>0\Rightarrow \log_4x>1\Rightarrow x>4$. Hence the domain is $x\in(4,\infty)$.

   Hint: Work from the outermost log inward; each argument must be positive.

8. 8.Given $\log_{10}2=0.30103$, find the number of digits in $2^{100}$.

   Number of digits $=\lfloor 100\log_{10}2\rfloor+1=\lfloor30.103\rfloor+1=30+1=\mathbf{31}$.

   Hint: Digits of $N$ equal $\lfloor\log_{10}N\rfloor+1$.

9. 9.If $\log_{10}7=0.8451$, how many digits does $7^{100}$ have?

   $\log_{10}7^{100}=100(0.8451)=84.51$. Number of digits $=\lfloor84.51\rfloor+1=\mathbf{85}$.

   Hint: Use the characteristic of $\log_{10}(7^{100})$.

10. 10.If the characteristic of $\log_{10}N$ is 3, what can you conclude about the number of digits in $N$ (for $N>1$)?

    Characteristic $=\lfloor\log_{10}N\rfloor=3$ means $10^3\le N<10^4$, so $N$ has exactly 4 digits.

    Hint: Characteristic plus 1 gives the digit count for numbers $\ge1$.

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