3. 10^3 = 1000, so 10 1000 = 3.

What is the value of 10 2 + 10 5?

1. 10 2 + 10 5 = 10(2 5) = 10 10 = 1.

1000. 10 x = 3 means x = 10^3 = 1000.

Given 10 2 = 0.3010, how many digits are there in 2^64?

20. 10(2^64) = 64 0.3010 = 19.264. The number of digits is 19.264 + 1 = 20.

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Quantitative Aptitude · Numbers & Simplification

Logarithms — Questions & Answers

Logarithm questions apply the product, quotient and power laws, plus change of base. Knowing that $\log_{10}2 \approx 0.3010$ and $\log_{10}3 \approx 0.4771$ lets you evaluate many expressions by hand.

5 solved questions
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Key formulas included

Key formulas

$\log_a(xy) = \log_a x + \log_a y$ , $\;\log_a\!\dfrac{x}{y} = \log_a x - \log_a y$
$\log_a(x^n) = n\log_a x$ , $\;\log_a a = 1$ , $\;\log_a 1 = 0$
Change of base: $\log_a b = \dfrac{\log_c b}{\log_c a} = \dfrac{1}{\log_b a}$

5 solved Logarithms questions

Try each question first — tap an option to check your answer, or open Learn step by step for a guided walkthrough.

01
Evaluate $\log_{10} 1000$ .
4 steps · reveal one at a time
1. Step 1: Understand Logarithms
  A logarithm asks "What power do I need to raise the base to, to get a certain number?". For example, $\log_b a = c$ means that $b^c = a$ . The base is $b$ , and $a$ is the number.
2. Step 2: Apply the Definition
  In our question, we need to evaluate $\log_{10} 1000$ . Here, the base is $10$ and the number is $1000$ . We are looking for the exponent, let's call it $x$ , such that $10^x = 1000$ .
3. Step 3: Find the Exponent
  Now, we need to determine what power of $10$ gives us $1000$ . We know that $10 \times 10 = 100$ , and $10 \times 10 \times 10 = 1000$ . Therefore, $10^3 = 1000$ .
4. Step 4: The answer
  So the answer is (A) 3. Since $10^3 = 1000$ , it means that $\log_{10} 1000 = 3$ by the definition of a logarithm.
02
Evaluate $\log_{2} 32$ .
4 steps · reveal one at a time
1. Step 1: Understand Logarithms
  A logarithm asks "what power do I need to raise the base to, to get a certain number?". So, $\log_b a = x$ is the same as saying $b^x = a$ .
2. Step 2: Rephrase the Question
  For $\log_2 32$ , our base is $2$ and the number is $32$ . We are looking for the power $x$ such that $2^x = 32$ . We need to find what power of $2$ gives $32$ .
3. Step 3: Calculate the Power
  Let's list powers of $2$ : $2^1 = 2$ , $2^2 = 4$ , $2^3 = 8$ , $2^4 = 16$ , and $2^5 = 32$ . So, the power we are looking for is $5$ .
4. Step 4: The answer
  So the answer is (C) 5. This is because $2$ raised to the power of $5$ equals $32$ , which is exactly what $\log_2 32$ asks.
03
What is the value of $\log_{10} 2 + \log_{10} 5$ ?
5 steps · reveal one at a time
1. Step 1: Recall log rule
  The expression involves the sum of two logarithms with the same base. Remember the product rule for logarithms: $\log_b x + \log_b y = \log_b (xy)$ . This rule will help us combine the terms.
2. Step 2: Apply the rule
  Using the product rule, we can combine $\log_{10} 2 + \log_{10} 5$ . Here, the base $b=10$ , $x=2$ , and $y=5$ . So, we get $\log_{10} (2 \times 5)$ . This simplifies the expression significantly.
3. Step 3: Simplify inside log
  Now, let's perform the multiplication inside the logarithm. $2 \times 5 = 10$ . So, our expression becomes $\log_{10} 10$ . We are getting closer to a final value.
4. Step 4: Evaluate the logarithm
  Finally, we need to evaluate $\log_{10} 10$ . Remember that $\log_b b = 1$ for any valid base $b$ . Since our base is 10 and the argument is also 10, the value is simply 1.
5. Step 5: The answer
  So the answer is (A) 1. The sum of the logarithms simplifies to $\log_{10} 10$ , which equals 1 based on the definition of a logarithm.
04
If $\log_{10} x = 3$ , find $x$ .
5 steps · reveal one at a time
1. Step 1: Understand the equation
  We are given the logarithmic equation $\log_{10} x = 3$ . Our task is to find the value of $x$ . This equation uses a logarithm with base 10.
2. Step 2: Recall logarithm definition
  The definition of a logarithm states that if $\log_b a = c$ , then this is equivalent to the exponential form $b^c = a$ . Here, $b$ is the base, $a$ is the argument, and $c$ is the exponent.
3. Step 3: Convert to exponential form
  Applying this definition to our equation $\log_{10} x = 3$ , we can identify the base $b=10$ , the exponent $c=3$ , and the argument $a=x$ . Therefore, the exponential form is $x = 10^3$ .
4. Step 4: Calculate the value
  Now, we simply need to calculate the value of $10^3$ . This means multiplying 10 by itself three times: $10 \times 10 \times 10 = 1000$ . So, $x = 1000$ .
5. Step 5: The answer
  So the answer is (B) 1000, because converting the logarithm $\log_{10} x = 3$ to its exponential form gives $x = 10^3$ , which equals 1000.
05
Given $\log_{10} 2 = 0.3010$ , how many digits are there in $2^{64}$ ?
5 steps · reveal one at a time
1. Step 1: Use logarithm properties
  To find the number of digits in $2^{64}$ , we can use base-10 logarithms. Recall the property that $\log_{10}(a^b) = b \log_{10} a$ . We are given $\log_{10} 2 = 0.3010$ .
2. Step 2: Calculate the logarithm
  Now, let's apply the property. We need to calculate $\log_{10}(2^{64})$ . This becomes $64 \times \log_{10} 2$ . Substitute the given value for $\log_{10} 2$ into this expression.
3. Step 3: Perform the multiplication
  Substitute the value: $64 \times 0.3010$ . Performing this multiplication gives us $19.264$ . So, $\log_{10}(2^{64}) = 19.264$ . This value is key to finding the number of digits.
4. Step 4: Find number of digits
  The number of digits in an integer $N$ is found by the formula $\lfloor \log_{10} N \rfloor + 1$ . Here, $N = 2^{64}$ and $\log_{10} N = 19.264$ . The floor function $\lfloor x \rfloor$ gives the greatest integer less than or equal to $x$ .
5. Step 5: The answer
  So the number of digits is $\lfloor 19.264 \rfloor + 1 = 19 + 1 = 20$ . So the answer is (C) 20. The number $2^{64}$ has 20 digits.

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