12B - Solving exponential equations:
We can use the index laws and observations from Section 12A to help solve exponential equations. The following examples demonstrate the process of solving exponential equations.
12B - Example 1: Solving simple exponential equations
Solve the following equations for \(x\).
(a) \(3^x=81\) (b) \(5 \times 2^x=160\) (c) \((\frac{1}{5})^x=125\) |
12B - Example 1: Video solution
12B - Example 1: Practice
Question 1: ABC Question 2: ABC 12B - Example 1: Solutions
Question 1: ABC Question 2: ABC |
12B - Example 2: Solving more complex exponential equations
Solve the equation \(3^{x+1} \times 3^{x-2}=81\) for \(x\).
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12B - Example 2: Video solution
12B - Example 2: Practice
Question 1: ABC Question 2: ABC 12B - Example 2: Solutions
Question 1: ABC Question 2: ABC |
12B - Example 3: Solving exponential equations with a quadratic structure
Solve the equation \(4^x-15 \times 2^x=16\) for \(x\).
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12B - Example 3: Video solution
12B - Example 3: Practice
Question 1: ABC Question 2: ABC 12B - Example 3: Solutions
Question 1: ABC Question 2: ABC |
12B - Example 4: Solving exponential equations with a quadratic structure (CAS)
Solve the equation \(5^x+5(5^{-x})=6\) for \(x\).
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12B - Example 4: Video solution
12B - Example 4: Practice
Question 1: ABC Question 2: ABC 12B - Example 4: Solutions
Question 1: ABC Question 2: ABC |
Solving exponential inequations:
When solving inequations involving exponentials, it is generally a good idea to sketch a simple graph to assist with the question.
12B - Example 5: Solving exponential inequations
Solve the following inequation for \(x\).
\[10^{\frac{x+1}{2}}>1000\]
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12B - Example 5: Video solution
12B - Example 5: Practice
Question 1: ABC Question 2: ABC 12B - Example 5: Solutions
Question 1: ABC Question 2: ABC |
12B - Example 6: Solving exponential inequations
Solve the following inequation for \(x\).
\[\left ( \frac{1}{2} \right )^x<4\]
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12B - Example 6: Video solution
12B - Example 6: Practice
Question 1: ABC Question 2: ABC 12B - Example 6: Solutions
Question 1: ABC Question 2: ABC |
Challenge: Most computers can't solve this, but you can!
Solve for all real numbers \(x\) for which: \[(x^2-7x+11)^{x^2-13x+42}=1\]
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Video solution:
Solutions:
This problem can be solve using index laws and considering different cases which give a value of 1.
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