Question

Solve (a) \(10^{\log x}=17\) (b) \(10^{2 \log x}=17\) (c) \(10^{x} 10^{2 x}=90\) (d) \(10^{2 x}=30\left(10^{2}\right)\)

Short answer

a) \(10^{\log x}=17\) b) \(10^{2 \log x}=17\) c) \(10^{x} 10^{2 x}=90\) d) \(10^{2 x}=30\left(10^{2}\right)\) Answer: The solutions for the given equations are as follows: a) \(x = 17\) b) \(x = \sqrt{17}, -\sqrt{17}\) c) \(x = \frac{\log(90)}{3}\) d) \(x = 1\)

Step-by-step solution

Use logarithm property

Recall that \(a^{\log_a x} = x\). This property can be applied here, since our base is 10. We have \(10^{\log x} = x\).

Substitute and solve for x

Substitute this property into the given equation: \(x = 17\). Therefore, the solution to this equation is \(x=17\). ##Equation (b) \(10^{2 \log x}=17\)##

Use logarithm property

Recall the property \(a^{n \log_a x } = x^n\). Apply this property here, with \(n=2\). We have \(10^{2\log x} = x^2\).

Substitute and solve for x

Substitute this property into the given equation: \(x^2 = 17\). Taking the square root of both sides, we obtain \(x = \pm\sqrt{17}\). Therefore, the solutions to this equation are \(x=\sqrt{17}, -\sqrt{17}\). ##Equation (c) \(10^{x} 10^{2 x}=90\)##

Use exponent property

Recall the property \(a^m \cdot a^n = a^{m+n}\). Apply this property here to combine the exponential terms: \(10^{x+2x} = 10^{3x}\).

Take logarithm of both sides

Take the logarithm base 10 of both sides: \(\log(10^{3x}) = \log(90)\).

Use logarithm property and solve for x

Recall the property \(\log_a (a^x) = x\). Apply this property to the left side of the equation: \(3x = \log(90)\). Divide both sides by 3: \(x = \frac{\log(90)}{3}\). Therefore, the solution to this equation is \(x = \frac{\log(90)}{3}\). ##Equation (d) \(10^{2 x}=30\left(10^{2}\right)\)##

Simplify the equation

Remove parentheses: \(10^{2x} = 30 \cdot 10^2\).

Divide both sides by 30

Divide both sides by 30 to isolate the exponential term: \(10^{2x} = 10^2\).

Take logarithm of both sides

Take the logarithm base 10 of both sides: \(\log(10^{2x}) = \log(10^2)\).

Use logarithm property and solve for x

Recall the property \(\log_a (a^x) = x\). Apply this property to both sides of the equation: \(2x = 2\). Divide both sides by 2: \(x = 1\). Therefore, the solution to this equation is \(x = 1\).

Key concepts

Exponential Functions

Exponential functions are foundational in mathematics and have wide applications, from calculating compound interest to modeling population growth. An exponential function is typically expressed in the form \(f(x) = a^x\), where \(a\) is a positive constant. In these equations, the variable \(x\) appears as an exponent.

When tackling exponential equations, it's essential to understand the rules of exponents, such as \(a^m \cdot a^n = a^{m+n}\). This principle allows us to multiply powers with the same base by simply adding their exponents. For example, in the expression \(10^x \cdot 10^{2x}\), we combine the terms to get \(10^{3x}\).

Understanding and applying these properties helps simplify and solve complex exponential equations effectively. By expressing problems in a unified base form, you can easily take logarithms and use additional properties to find the solutions.

Logarithm Properties

Logarithms are the inverse functions of exponentials and play a crucial role in converting multiplicative operations into additive ones. The basic idea of a logarithm is to find the power to which a number, called the base, must be raised to yield a specific value. For example, if \(10^x = 100\), then \(\log_{10}(100) = x = 2\).

Logarithmic properties are vital tools for solving equations involving exponents. Some of the fundamental properties include:

• \(\log_a(xy) = \log_a x + \log_a y\)
• \(\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y\)
• \(\log_a(x^n) = n \cdot \log_a x\)

When you encounter an equation like \(10^{2\log x} = 17\), employing these properties allows you to simplify and transform the expression into a more manageable form, such as \(x^2 = 17\). This transformation enables you to solve for \(x\) by taking the square root of both sides.

Mathematical Problem Solving

Problem solving in mathematics involves a structured approach to tackling equations, using a keen insight into properties and operations, and requires critical thinking. To solve mathematical problems efficiently, it's important to follow certain steps:

• *Understand the problem*: Identify the type of equation and relevant mathematical properties that can be applied.
• *Strategy formulation*: Plan an approach based on known principles, such as applying exponential or logarithmic properties to simplify the problem.
• *Execution*: Carry out the steps thoughtfully, checking your work as you progress. This might include taking logarithms, using base properties, or other simplification techniques.
• *Verification*: Always check your solution by substituting back into the original problem to confirm correctness.

By integrating these steps, mathematicians and students alike can tackle complex problems systematically. For instance, solving \(10^{x} \cdot 10^{2x} = 90\) requires the use of exponent addition, followed by logarithms to isolate the variable, ensuring a precise and correct result.