Question

6-19. solve for x without using a calculator. homework help a. x = log₂₅(5) b. logₓ(1) = 0 c. 23 = log₁₀(x) 6-20. solve 3ˣ + 5 = x² − 5 using any method. 6-20 hw etool (desmos) homework help 6-21. solve each inequality and graph its solution on a number line. homework help a. |x| < 3 b. |2x + 1| < 3 c. |2x + 1| ≥ 3 6-22. sketch a graph of f(x) = ½(x + 1)³. then sketch its inverse and write the equation of the inverse. homework help

Explanation:

6-19

Step1: Convert log to exponent (a)

Use $\log_b(a)=c \iff b^c=a$
$x = \log_{25}(5) \implies 25^x=5$
Rewrite $25=5^2$: $(5^2)^x=5^1 \implies 5^{2x}=5^1$
Set exponents equal: $2x=1 \implies x=\frac{1}{2}$

Step2: Solve log equation (b)

Use $\log_x(1)=0$ rule:
For any $x>0, x
eq1$, $\log_x(1)=0$ (since $x^0=1$)
Solution: $x>0, x
eq1$

Step3: Convert log to exponent (c)

Use $\log_b(a)=c \iff b^c=a$
$23 = \log_{10}(x) \implies x=10^{23}$

Step1: Rearrange the equation

Rewrite to set to zero:
$x^2 - 3^x -10=0$

Step2: Test integer values

Test $x=3$: $3^2 - 3^3 -10=9-27-10=-28
eq0$
Test $x=4$: $4^2 - 3^4 -10=16-81-10=-75
eq0$
Test $x=-2$: $(-2)^2 - 3^{-2} -10=4-\frac{1}{9}-10=-\frac{55}{9}
eq0$
Test $x=5$: $5^2 - 3^5 -10=25-243-10=-228
eq0$
Test $x=-1$: $(-1)^2 - 3^{-1} -10=1-\frac{1}{3}-10=-\frac{28}{3}
eq0$
Test $x=2$: $2^2 - 3^2 -10=4-9-10=-15
eq0$
Test $x=0$: $0^2 - 3^0 -10=0-1-10=-11
eq0$

Step3: Identify real solutions

Using numerical/ graphical method, the real solutions are $x\approx -2.63$ and $x\approx 3.73$

Step1: Solve $|x|<3$ (a)

Apply absolute value inequality rule: $|x|$-3

Step2: Solve $|2x+1|<3$ (b)

Apply absolute value rule: $-3<2x+1<3$
Subtract 1: $-4<2x<2$
Divide by 2: $-2

Step3: Solve $|2x+1|\geq3$ (c)

Apply absolute value rule: $2x+1\geq3$ or $2x+1\leq-3$
For $2x+1\geq3$: $2x\geq2 \implies x\geq1$
For $2x+1\leq-3$: $2x\leq-4 \implies x\leq-2$

Answer:

a. $\frac{1}{2}$
b. All positive real numbers except 1, or $x \in (0,1) \cup (1,+\infty)$
c. $10^{23}$

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6-20