Question Solve exponential equations, analyze errors, and explain logarithm laws. Equations: $7=(3)^{x}-2$ , $(16)^{x}=(2)^{x-2}$ . Error analysis: $5^{3x+2}=25^{x-8}$ , $\left(\frac{1}{6}\right)^{5x}=32^{x+8}$ . Logarithm laws: relation to exponential numbers, 7 laws.

Studdy Solution

STEP 1

Assumptions
1. We are solving exponential equations where the base and the exponent are related.
2. The properties of exponents and logarithms will be used to solve the equations.
3. For the error analysis, we will identify and correct the mistakes made in the given solutions.

STEP 2

Solve the first equation $7=(3)^{x}-2$ .

STEP 3

Add 2 to both sides of the equation to isolate the exponential term.
$7 + 2 = (3)^{x}-2 + 2$

STEP 4

Calculate the sum on the left side of the equation.
$9 = (3)^{x}$

STEP 5

Recognize that $9 = 3^2$ and write the equation with the same base.
$(3)^{2} = (3)^{x}$

STEP 6

Since the bases are the same, set the exponents equal to each other.
$2 = x$

STEP 7

Check the solution by substituting $x = 2$ back into the original equation.
$7 = (3)^{2} - 2$

STEP 8

Calculate the right side of the equation.
$7 = 9 - 2$

STEP 9

Verify that the left side equals the right side after simplification.
$7 = 7$

STEP 10

The solution to the first equation is $x = 2$ .

STEP 11

Solve the second equation $(16)^{x}=(2)^{x-2}$ .

STEP 12

Recognize that $16 = 2^4$ and write the equation with the same base.
$(2^4)^{x} = (2)^{x-2}$

STEP 13

Apply the power of a power rule to the left side of the equation.
$2^{4x} = 2^{x-2}$

STEP 14

Since the bases are the same, set the exponents equal to each other.
$4x = x - 2$

STEP 15

Subtract $x$ from both sides of the equation to solve for $x$ .
$4x - x = x - 2 - x$

STEP 16

Simplify the equation.
$3x = -2$

STEP 17

Divide both sides by 3 to solve for $x$ .
$x = \frac{-2}{3}$

STEP 18

Check the solution by substituting $x = -\frac{2}{3}$ back into the original equation.
$(16)^{-\frac{2}{3}} = (2)^{-\frac{2}{3}-2}$

STEP 19

Simplify the right side of the equation.
$(16)^{-\frac{2}{3}} = (2)^{-\frac{8}{3}}$

STEP 20

Recognize that $(16)^{-\frac{2}{3}} = (2^4)^{-\frac{2}{3}}$ and simplify.
$(2^4)^{-\frac{2}{3}} = (2)^{-\frac{8}{3}}$

STEP 21

Apply the power of a power rule to the left side of the equation.
$2^{-\frac{8}{3}} = 2^{-\frac{8}{3}}$

STEP 22

Verify that the left side equals the right side after simplification.
$2^{-\frac{8}{3}} = 2^{-\frac{8}{3}}$

STEP 23

The solution to the second equation is $x = -\frac{2}{3}$ .

STEP 24

Error analysis for the equation $5^{3x+2} = 25^{x-8}$ .

STEP 25

Identify the mistake: The error is in assuming that the exponents of the bases 5 and 25 can be set equal to each other directly. However, 25 is actually $5^2$ , and this needs to be taken into account.

STEP 26

Correct the equation by writing 25 as $5^2$ .
$5^{3x+2} = (5^2)^{x-8}$

STEP 27

Apply the power of a power rule to the right side of the equation.
$5^{3x+2} = 5^{2(x-8)}$

STEP 28

Since the bases are the same, set the exponents equal to each other.
$3x+2 = 2(x-8)$

STEP 29

Expand the right side of the equation.
$3x+2 = 2x - 16$

STEP 30

Solve for $x$ by subtracting $2x$ from both sides and adding 16 to both sides.
$x = -18$
The corrected solution for equation 19 is $x = -18$ .

STEP 31

Error analysis for the equation $\left(\frac{1}{6}\right)^{5x} = 32^{x+8}$ .

STEP 32

Identify the mistake: The error is in assuming that $\left(\frac{1}{6}\right)^{5x}$ can be written as $(2^3)^{5x}$ . However, $\frac{1}{6}$ is actually $2^{-1} \cdot 3^{-1}$ , and this needs to be taken into account.

STEP 33

Correct the equation by writing $\frac{1}{6}$ as $2^{-1} \cdot 3^{-1}$ and $32$ as $2^5$ .
$\left(2^{-1} \cdot 3^{-1}\right)^{5x} = (2^5)^{x+8}$

STEP 34

Apply the power of a power rule to both sides of the equation.
$2^{-5x} \cdot 3^{-5x} = 2^{5(x+8)}$

STEP 35

Since the bases are different (2 and 3), the equation cannot be simplified by setting the exponents equal to each other. This indicates that the original approach was incorrect.

STEP 36

Recognize that the equation has no solution because the left side involves two different bases being multiplied, while the right side has only one base. The exponents of different bases cannot be equated.
The corrected conclusion for equation 20 is that there is no solution.

STEP 37

Explain how logarithm is related to exponential numbers.

STEP 38

A logarithm is the inverse operation to exponentiation. If $a^b = c$ , then the logarithm of $c$ with base $a$ is $b$ , written as $\log_a(c) = b$ .

STEP 39

Write the 7 laws of logarithms.
1. $\log_b(mn) = \log_b(m) + \log_b(n)$ (Product Rule)
2. $\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)$ (Quotient Rule)
3. $\log_b(m^n) = n \log_b(m)$ (Power Rule)
4. $\log_b(b) = 1$ (Base Rule)
5. $\log_b(1) = 0$ (Identity Rule)
6. $\log_b(m) = \frac{\log_c(m)}{\log_c(b)}$ (Change of Base Rule)
7. $b^{\log_b(m)} = m$ (Inverse Rule)

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