The Sum of Logarithms - Examples, Exercises and Solutions

Understanding The Sum of Logarithms

Complete explanation with examples

Addition of Logarithms

The definition of a logarithm is:


logax=blog_a⁡x=b
X=abX=a^b

Where:
aa is the base of the exponent
XX is what appears inside the log, can also appear in parentheses
bb is the exponent we raise the log base to in order to get the number that appears inside the log.

Adding logarithms with the same base is based on the following rule:


logax+logay=loga(xy)log_a⁡x+log_a⁡y=log_a⁡(x\cdot y)

Visual explanation of logarithmic rules showing log(x·y) equals log(x) plus log(y), and log(x/y) equals log(x) minus log(y), with arrows connecting each part for clarity.

Adding logarithms with different bases is done by changing the base of the log using the following rule:

logaX=logbase we want to change toXlogbase we want to change toalog_aX=\frac{log_{base~we~want~to~change~to}X}{log_{base~we~want~to~change~to}a}

Logarithmic change of base formula illustrated: log base b of a equals log base x of a divided by log base x of b, with arrows showing transformation from original form.

Detailed explanation

Practice The Sum of Logarithms

Test your knowledge with 6 quizzes

\( \log4x+\log2-\log9=\log_24 \)

?=x

Examples with solutions for The Sum of Logarithms

Step-by-step solutions included
Exercise #1

log103+log104= \log_{10}3+\log_{10}4=

Step-by-Step Solution

To solve this problem, we will use the property of logarithms that allows us to combine the sum of two logarithms:

  • Step 1: Identify the formula. We use the property logb(x)+logb(y)=logb(xy)\log_b(x) + \log_b(y) = \log_b(x \cdot y) where both logarithms must have the same base.
  • Step 2: Recognize the base. Here, both logarithms are in base 10: log103\log_{10}3 and log104\log_{10}4.
  • Step 3: Apply the property. Add the two logarithms using the formula: log103+log104=log10(34)\log_{10}3 + \log_{10}4 = \log_{10}(3 \cdot 4).
  • Step 4: Perform the multiplication. Compute 343 \cdot 4 to get 12.
  • Step 5: Express the result as a single logarithm: log1012\log_{10}12.

Therefore, the expression log103+log104\log_{10}3 + \log_{10}4 simplifies to log1012\log_{10}12.

Answer:

log1012 \log_{10}12

Video Solution
Exercise #2

log24+log25= \log_24+\log_25=

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the given expression as log24+log25 \log_2 4 + \log_2 5 .
  • Step 2: Use the sum of logarithms rule to simplify the expression.
  • Step 3: Calculate the product and express the result.

Let's work through each step:

Step 1: We have log24+log25 \log_2 4 + \log_2 5 as our expression.

Step 2: Apply the sum of logarithms formula:

log24+log25=log2(45) \log_2 4 + \log_2 5 = \log_2 (4 \cdot 5)

Step 3: Calculate the product:

4×5=20 4 \times 5 = 20

Thus, log2(45)=log220 \log_2 (4 \cdot 5) = \log_2 20 .

Therefore, the solution to the problem is log220 \log_2 20 .

Answer:

log220 \log_220

Video Solution
Exercise #3

log974+log912= \log_974+\log_9\frac{1}{2}=

Step-by-Step Solution

To solve this problem, we'll apply the following steps:

  • Step 1: Identify given logarithms and their base.
  • Step 2: Employ the sum of logarithms property to combine the terms.
  • Step 3: Calculate the resulting argument of the logarithm.

Now, let's work through each step:

Step 1: We have two logarithms: log974\log_9 74 and log912\log_9 \frac{1}{2}, sharing the base of 99.

Step 2: Since the bases are the same, we use the sum property of logarithms:

log974+log912=log9(74×12)\log_9 74 + \log_9 \frac{1}{2} = \log_9 (74 \times \frac{1}{2}).

Step 3: Calculate the product 74×1274 \times \frac{1}{2}:

74×12=3774 \times \frac{1}{2} = 37.

So, we have:

log9(74×12)=log937\log_9 (74 \times \frac{1}{2}) = \log_9 37.

Therefore, the solution to the problem is log937\log_9 37.

Answer:

log937 \log_937

Video Solution
Exercise #4

2log82+log83= 2\log_82+\log_83=

Step-by-Step Solution

2log82=log822=log84 2\log_82=\log_82^2=\log_84

2log82+log83=log84+log83= 2\log_82+\log_83=\log_84+\log_83=

log843=log812 \log_84\cdot3=\log_812

Answer:

log812 \log_812

Video Solution
Exercise #5

3log49+8log413= 3\log_49+8\log_4\frac{1}{3}=

Step-by-Step Solution

Where:

3log49=log493=log4729 3\log_49=\log_49^3=\log_4729

y

8log413=log4(13)8= 8\log_4\frac{1}{3}=\log_4\left(\frac{1}{3}\right)^8=

log4138=log416561 \log_4\frac{1}{3^8}=\log_4\frac{1}{6561}

Therefore

3log49+8log413= 3\log_49+8\log_4\frac{1}{3}=

log4729+log416561 \log_4729+\log_4\frac{1}{6561}

logax+logay=logaxy \log_ax+\log_ay=\log_axy

(72916561)=log419 \left(729\cdot\frac{1}{6561}\right)=\log_4\frac{1}{9}

log491=log49 \log_49^{-1}=-\log_49

Answer:

log49 -\log_49

Video Solution

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