Example Question - exponential equations
Here are examples of questions we've helped users solve.
Finding the Value of n in an Exponential Equation
<p>Given the equation:</p> <p>(x<sup>n</sup>)<sup>3</sup> = \frac{x<sup>18</sup>}{x<sup>-6</sup>}</p> <p>We can simplify the right side:</p> <p>\frac{x<sup>18</sup>}{x<sup>-6</sup>} = x<sup>18 - (-6)</sup> = x<sup>18 + 6</sup> = x<sup>24</sup></p> <p>Now we have:</p> <p>(x<sup>n</sup>)<sup>3</sup> = x<sup>24</sup></p> <p>Using the power of a power rule, the left side becomes:</p> <p>x<sup>3n</sup> = x<sup>24</sup></p> <p>Since the bases are the same, we can set the exponents equal to each other:</p> <p>3n = 24</p> <p>Now solve for n:</p> <p>n = \frac{24}{3} = 8</p> <p>The value of n is 8.</p>
Complex Logarithmic Expression Simplification
<p>\[\text{Given expression: } \frac{\log 343}{1 + \frac{1}{2}\log(49) - \frac{1}{3}\log\left(\frac{1}{125}\right)}\]</p> <p>\[= \frac{\log 7^3}{1 + \frac{1}{2}\log(7^2) - \frac{1}{3}\log\left(5^{-3}\right)}\]</p> <p>\[= \frac{3\log 7}{1 + \frac{1}{2}\cdot 2 \log 7 - \frac{1}{3} \cdot (-3) \log 5}\]</p> <p>\[= \frac{3\log 7}{1 + \log 7 + \log 5}\]</p> <p>\[= \frac{3\log 7}{\log 7 + \log 5 + \log 10}\]</p> <p>\[= \frac{3\log 7}{\log(7 \cdot 5 \cdot 10)}\]</p> <p>\[= \frac{3\log 7}{\log 350}\]</p> <p>\[\text{Since } \log_a b = \frac{1}{\log_b a}, \text{ swap the logs:}\]</p> <p>\[= \frac{3}{\log_{7} 350}\]</p> <p>\[= \frac{3}{\log_{7} 7^2 + \log_{7} 5}\]</p> <p>\[= \frac{3}{2 + \log_{7} 5}\]</p> <p>\[\text{Without information on the value of } \log_{7} 5, \text{ this is the simplest form.}\]</p>
Solving Exponential Equations with Same Base
To solve the equation \( 8^{x+1} = 2^{5-x} \) using a common base, we first need to express both sides of the equation with the same base. We know that 8 is equivalent to \( 2^3 \). Thus, we can rewrite \( 8^{x+1} \) as \( (2^3)^{x+1} \), which simplifies to \( 2^{3(x+1)} \). Now the equation becomes: \( 2^{3(x+1)} = 2^{5-x} \) Since the bases are now the same, we can set the exponents equal to each other: \( 3(x+1) = 5 - x \) Next, we distribute the 3 on the left side: \( 3x + 3 = 5 - x \) Now bring all x terms to one side and constants to the other side: \( 3x + x = 5 - 3 \) Combine like terms: \( 4x = 2 \) Finally, divide by 4 to get x alone: \( x = \frac{2}{4} \) Simplify the fraction: \( x = \frac{1}{2} \) So the solution is x = 1/2.
Solving Exponential Equations with Equal Bases
To solve the equation e^(2x) = e^(3x-1), we will use the property that if two exponents with the same base are equal, then their exponents must be equal as well. This means we can set the exponents equal to each other and solve for x. 2x = 3x - 1 Now, let's solve for x: 2x - 3x = -1 -x = -1 Multiply both sides by -1 to isolate x: x = 1 So, the solution to the equation is x = 1.
Converting Exponential to Logarithmic Form
The image shows a handwritten question asking to convert the form of the following equation, specifically from exponential form to logarithmic form, and vice versa. The given equation is: 7^x = 2401 To convert from exponential form to logarithmic form, you apply the basic definition of logarithms. If a^b = c, then log_a(c) = b. Applying this to the given equation, you get: log_7(2401) = x. To find the value of x, you need to determine what power of 7 gives you 2401. 2401 is 7 raised to the fourth power since 7 * 7 * 7 * 7 = 2401. Therefore, the logarithmic form of the equation is: log_7(2401) = 4. Hence, x = 4.
Solving Transcendental Equations Numerically
The image contains an equation: \[ e^x + 7x + 11 = 20 \] To solve this equation for \( x \), follow these steps: 1. First, move the constant term (20) on the right to the left side of the equation: \[ e^x + 7x + 11 - 20 = 0 \] 2. Simplify the left-hand side of the equation: \[ e^x + 7x - 9 = 0 \] 3. This equation cannot be solved algebraically using elementary methods because it is a transcendental equation (it involves both an exponential and a linear term in \( x \)). Therefore, the solution for \( x \) generally involves numerical methods or iterative techniques like the Newton-Raphson method. You can use a graphing utility or calculator to determine the root approximately. Another option is to use software capable of solving such equations, such as Mathematica, MATLAB, or Python's SciPy library. Do you need an approximate solution using numerical methods, or is this sufficient for your needs?
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