Logarithms can be intimidating, but solving a logarithm is much easier once you realize that logarithms are just a different way to write exponential equations. Once the logarithms are rewritten in a more familiar form, you should be able to solve them as a standard exponential equation.
Steps
Learn to Express Logarithmic Equations Exponentially
Step 1. Learn the Definition of Logarithm
Before you can solve logarithms, you need to understand that a logarithm is essentially a different way to write exponential equations. Its precise definition is as follows:
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y = logb (x)
If and only if: by = x
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Note that b is the base of the logarithm. It must also be true that:
- b> 0
- b is not equal to 1
- In the same equation, y is the exponent and x is the exponential expression to which the logarithm is equaled.
Step 2. Analyze the equation
When you are faced with a logarithmic problem, identify the base (b), the exponent (y), and the exponential expression (x).
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Example:
5 = log4(1024)
- b = 4
- y = 5
- x = 1024
Solve Logarithms Step 3 Step 3. Move the exponential expression to one side of the equation
Place the value of your exponential expression, x, on one side of the equal sign.
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Example: 1024 = ?
Solve Logarithms Step 4 Step 4. Apply the exponent to the base
The value of your base, b, must be multiplied by itself the number of times indicated by the exponent, y.
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Example:
4 * 4 * 4 * 4 * 4 = ?
This could also be written as: 45
Solve Logarithms Step 5 Step 5. Rewrite your final answer
You should now be able to rewrite your logarithm as an exponential expression. Check that your expression is correct by making sure that the members on both sides of the equals are equivalent.
Example: 45 = 1024
Method 1 of 3: Method 1: Solve for X
Solve Logarithms Step 6 Step 1. Isolate the logarithm
Use the inverse operation to bring all the parts that are not logarimic to the other side of the equation.
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Example:
log3(x + 5) + 6 = 10
- log3(x + 5) + 6 - 6 = 10 - 6
- log3(x + 5) = 4
Solve Logarithms Step 7 Step 2. Rewrite the equation in exponential form
Using what you know about the relationship between logarithmic equations and exponentials, break down the logarithm and rewrite the equation in exponential form, which is easier to solve.
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Example:
log3(x + 5) = 4
- Comparing this equation with the definition [ y = logb (x)], you can conclude that: y = 4; b = 3; x = x + 5
- Rewrite the equation so that: by = x
- 34 = x + 5
Solve Logarithms Step 8 Step 3. Solve for x
With the simplified problem to an exponential, you should be able to solve it as you would solve an exponential.
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Example:
34 = x + 5
- 3 * 3 * 3 * 3 = x + 5
- 81 = x + 5
- 81 - 5 = x + 5 - 5
- 76 = x
Solve Logarithms Step 9 Step 4. Write your final answer
The solution you find solving for x is the solution of your original logarithm.
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Example:
x = 76
Method 2 of 3: Method 2: Solve for X Using the Logarithmic Product Rule
Solve Logarithms Step 10 Step 1. Learn the product rule
The first property of logarithms, called the "product rule," says that the logarithm of a product is the sum of the logarithms of the various factors. Writing it through an equation:
- logb(m * n) = logb(m) + logb(n)
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Also note that the following conditions must be met:
- m> 0
- n> 0
Solve Logarithms Step 11 Step 2. Isolate the logarithm from one side of the equation
Use the operations of the inverai to bring all the parts containing logarithms on one side of the equation and all the rest on the other.
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Example:
log4(x + 6) = 2 - log4(x)
- log4(x + 6) + log4(x) = 2 - log4(x) + log4(x)
- log4(x + 6) + log4(x) = 2
Solve Logarithms Step 12 Step 3. Apply the product rule
If there are two logarithms that are added together within the equation, you can use the logarithm rules to combine them together and transform them into one. Note that this rule only applies if the two logarithms have the same base
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Example:
log4(x + 6) + log4(x) = 2
- log4[(x + 6) * x] = 2
- log4(x2 + 6x) = 2
Solve Logarithms Step 13 Step 4. Rewrite the equation in exponential form
Remember that the logarithm is just another way to write the exponential. Rewrite the equation in a solvable form
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Example:
log4(x2 + 6x) = 2
- Compare this equation with the definition [ y = logb (x)], then conclude that: y = 2; b = 4; x = x2 + 6x
- Rewrite the equation so that: by = x
- 42 = x2 + 6x
Solve Logarithms Step 14 Step 5. Solve for x
Now that the equation has become a standard exponential, use your knowledge of exponential equations to solve for x as you normally would.
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Example:
42 = x2 + 6x
- 4 * 4 = x2 + 6x
- 16 = x2 + 6x
- 16 - 16 = x2 + 6x - 16
- 0 = x2 + 6x - 16
- 0 = (x - 2) * (x + 8)
- x = 2; x = -8
Solve Logarithms Step 15 Step 6. Write your answer
At this point you should know the solution of the equation, which corresponds to that of the starting equation.
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Example:
x = 2
- Note that you cannot have a negative solution for logarithms, so you discard the solution x = - 8.
Method 3 of 3: Method 3: Solve for X Using the Logarithmic Quotient Rule
Solve Logarithms Step 16 Step 1. Learn the quotient rule
According to the second property of logarithms, called the "quotient rule," the logarithm of a quotient can be rewritten as the difference between the logarithm of the numerator and the logarithm of the denominator. Writing it as an equation:
- logb(m / n) = logb(m) - logb(n)
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Also note that the following conditions must be met:
- m> 0
- n> 0
Solve Logarithms Step 17 Step 2. Isolate the logarithm from one side of the equation
Before you can solve the logarithm, you have to move all the logarithms to one side of the equation. Everything else should be moved to the other member. Use inverse operations to accomplish this.
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Example:
log3(x + 6) = 2 + log3(x - 2)
- log3(x + 6) - log3(x - 2) = 2 + log3(x - 2) - log3(x - 2)
- log3(x + 6) - log3(x - 2) = 2
Solve Logarithms Step 18 Step 3. Apply the quotient rule
If there is a difference between two logarithms having the same base within the equation, you must use the rule of quotients to rewrite the logarithms as one.
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Example:
log3(x + 6) - log3(x - 2) = 2
log3[(x + 6) / (x - 2)] = 2
Solve Logarithms Step 19 Step 4. Rewrite the equation in exponential form
Remember that the logarithm is just another way to write the exponential. Rewrite the equation in a solvable form.
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Example:
log3[(x + 6) / (x - 2)] = 2
- Comparing this equation to the definition [ y = logb (x)], you can conclude that: y = 2; b = 3; x = (x + 6) / (x - 2)
- Rewrite the equation so that: by = x
- 32 = (x + 6) / (x - 2)
Solve Logarithms Step 20 Step 5. Solve for x
With the equation now in exponential form, you should be able to solve for x as you normally would.
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Example:
32 = (x + 6) / (x - 2)
- 3 * 3 = (x + 6) / (x - 2)
- 9 = (x + 6) / (x - 2)
- 9 * (x - 2) = [(x + 6) / (x - 2)] * (x - 2)
- 9x - 18 = x + 6
- 9x - x - 18 + 18 = x - x + 6 + 18
- 8x = 24
- 8x / 8 = 24/8
- x = 3
Solve Logarithms Step 21 Step 6. Write your final solution
Go back and double check your steps. Once you are sure you have the correct solution, write it down.
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Example:
x = 3
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