Solve the following two math problems involving exponents and logarithms:

Question

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here are the solutions to the math problems.

1. Solve for $x$ :

4^{x+1} \times 2^{2x-3} = 512

Express all numbers with a base of 2.
$(2^2)^{x+1} \times 2^{2x-3} = 2^9$
Apply the power rule $(a^m)^n = a^{mn}$ .
$2^{2(x+1)} \times 2^{2x-3} = 2^9$ $2^{2x+2} \times 2^{2x-3} = 2^9$
Apply the product rule $a^m \times a^n = a^{m+n}$ .
$2^{(2x+2) + (2x-3)} = 2^9$ $2^{4x-1} = 2^9$
Equate the exponents since the bases are the same.
$4x-1 = 9$
Solve for $x$ .
$4x = 9+1$ $4x = 10$ $x = \frac{10}{4}$ $x = \frac{5}{2}$ $x = 2.5$
The value of $x$ is $\boxed{2.5}$ .

2. Use logarithms to evaluate:
$\frac{2.735 \times 0.0846}{0.693}$
Let $N = \frac{2.735 \times 0.0846}{0.693}$ .
Take the logarithm (base 10) of both sides.
$\log N = \log \left( \frac{2.735 \times 0.0846}{0.693} \right)$
Apply logarithm properties $\log(AB) = \log A + \log B$ and $\log(\frac{A}{B}) = \log A - \log B$ .
$\log N = \log(2.735) + \log(0.0846) - \log(0.693)$
Find the logarithms of each number.
$\log(2.735) \approx 0.4376$ $\log(0.0846) \approx \bar{2}.9274 \quad \text{or} \quad -1.0726$ $\log(0.693) \approx \bar{1}.8407 \quad \text{or} \quad -0.1593$
Using characteristic and mantissa:
$\log(2.735) = 0.4376$ $\log(0.0846) = 2.9274$ $\log(0.693) = 1.8407$
Perform the addition and subtraction.
$\log N = 0.4376 + \bar{2}.9274 - \bar{1}.8407$ $\log N = (0.4376 - 2 + 0.9274) - (-1 + 0.8407)$ $\log N = (0.4376 + 0.9274 - 2) - (0.8407 - 1)$ $\log N = 1.3650 - 2 - 0.8407 + 1$ $\log N = 0.5243 - 1$ $\log N = \bar{1}.5243$
Find the antilogarithm.
$N = 10^{\bar{1}.5243}$ $N = 10^{-1} \times 10^{0.5243}$ $N \approx 0.1 \times 3.344$ $N \approx 0.3344$
The evaluated value is approximately $\boxed{0.3344}$ .

3. The volume of a cylindrical piece of wood is $2187 \text{ cm}^3$ , given that the area of the base is $243 \text{ cm}^2$ , use the laws of indices and a base of 3 to find the height of the piece of wood leaving the answer in index form.
Recall the formula for the volume of a cylinder.
$V = A_{base} \times h$
Substitute the given values.
$2187 \text{ cm}^3 = 243 \text{ cm}^2 \times h$
Solve for $h$ .
$h = \frac{2187}{243} \text{ cm}$
Express 2187 and 243 as powers of 3.
$3^1 = 3$ $3^2 = 9$ $3^3 = 27$ $3^4 = 81$ $3^5 = 243$ $3^6 = 729$ $3^7 = 2187$
So, $2187 = 3^7$ and $243 = 3^5$ .
Substitute the index forms into the equation for $h$ .
$h = \frac{3^7}{3^5}$
Apply the division rule for indices $a^m / a^n = a^{m-n}$ .
$h = 3^{7-5}$ $h = 3^2 \text{ cm}$
The height of the piece of wood in index form is $\boxed{3^2 \text{ cm}}$ .

3. The length of a rectangular hall is $(x+4)$ m and the width is $(x-3)$ m. If its area is $77 \text{ m}^2$ , find its dimensions.
Write the formula for the area of a rectangle.
$\text{Area} = \text{Length} \times \text{Width}$
Substitute the given expressions for length, width, and area.
$77 = (x+4)(x-3)$
Expand the right side of the equation.
$77 = x^2 - 3x + 4x - 12$ $77 = x^2 + x - 12$
Rearrange the equation into a standard quadratic form $ax^2 + bx + c = 0$ .
$x^2 + x - 12 - 77 = 0$ $x^2 + x - 89 = 0$
Solve the quadratic equation for $x$ .
We can use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Here, $a=1$ , $b=1$ , $c=-89$ .
$x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-89)}}{2(1)}$ $x = \frac{-1 \pm \sqrt{1 + 356}}{2}$ $x = \frac{-1 \pm \sqrt{357}}{2}$ $x \approx \frac{-1 \pm 18.894}{2}$
Two possible values for $x$ :
$x_1 = \frac{-1 + 18.894}{2} = \frac{17.894}{2} \approx 8.947$ $x_2 = \frac{-1 - 18.894}{2} = \frac{-19.894}{2} \approx -9.947$
Choose the valid value for $x$ .
Since dimensions (length and width) cannot be negative, $x-3$ must be positive. If $x = -9.947$ , then $x-3 = -12.947$ , which is not possible. Therefore, we use $x \approx 8.947$ .
Calculate the dimensions using $x \approx 8.947$ .
$\text{Length} = x+4 = 8.947 + 4 = 12.947 \text{ m}$ $\text{Width} = x-3 = 8.947 - 3 = 5.947 \text{ m}$
The dimensions of the hall are approximately $\boxed{\text{Length} = *12.95 \text{ m}*, \text{Width} = *5.95 \text{ m}}$ .

4. List all the composite numbers from 81 to 100.
A composite number is a positive integer that has at least one divisor other than 1 and itself. In other words, it is not a prime number.
Numbers from 81 to 100:
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100.
Identify prime numbers in this range (numbers only divisible by 1 and themselves).
- 83 (divisors: 1, 83)
- 89 (divisors: 1, 89)
- 97 (divisors: 1, 97)
All other numbers in the list are composite.
- 81 ( $9 \times 9$ )
- 82 ( $2 \times 41$ )
- 84 ( $2 \times 42$ )
- 85 ( $5 \times 17$ )
- 86 ( $2 \times 43$ )
- 87 ( $3 \times 29$ )
- 88 ( $2 \times 44$ )
- 90 ( $9 \times 10$ )
- 91 ( $7 \times 13$ )
- 92 ( $2 \times 46$ )
- 93 ( $3 \times 31$ )
- 94 ( $2 \times 47$ )
- 95 ( $5 \times 19$ )
- 96 ( $2 \times 48$ )
- 98 ( $2 \times 49$ )
- 99 ( $9 \times 11$ )
- 100 ( $10 \times 10$ )
The composite numbers from 81 to 100 are:
$\boxed{81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100}$ .

Solve the following two math problems involving exponents and logarithms:

Step-by-step solution

Express all numbers with a base of 2.

Apply the power rule $(a^m)^n = a^{mn}$ .

Apply the product rule $a^m \times a^n = a^{m+n}$ .

Equate the exponents since the bases are the same.

Solve for $x$ .

Take the logarithm (base 10) of both sides.

Apply logarithm properties $\log(AB) = \log A + \log B$ and $\log(\frac{A}{B}) = \log A - \log B$ .

Find the logarithms of each number.

Perform the addition and subtraction.

Find the antilogarithm.

Recall the formula for the volume of a cylinder.

Substitute the given values.

Solve for $h$ .

Express 2187 and 243 as powers of 3.

Substitute the index forms into the equation for $h$ .

Apply the division rule for indices $a^m / a^n = a^{m-n}$ .

Write the formula for the area of a rectangle.

Substitute the given expressions for length, width, and area.

Expand the right side of the equation.

Rearrange the equation into a standard quadratic form $ax^2 + bx + c = 0$ .

Solve the quadratic equation for $x$ .

Choose the valid value for $x$ .

Calculate the dimensions using $x \approx 8.947$ .

Identify prime numbers in this range (numbers only divisible by 1 and themselves).

All other numbers in the list are composite.

Still stuck on this one?

More Mathematics Questions

Solve the following two math problems involving exponents and logarithms:

Step-by-step solution

Express all numbers with a base of 2.

Apply the power rule (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Apply the product rule am×an=am+na^m \times a^n = a^{m+n}am×an=am+n.

Equate the exponents since the bases are the same.

Solve for xxx.

Take the logarithm (base 10) of both sides.

Apply logarithm properties log⁡(AB)=log⁡A+log⁡B\log(AB) = \log A + \log Blog(AB)=logA+logB and log⁡(AB)=log⁡A−log⁡B\log(\frac{A}{B}) = \log A - \log Blog(BA​)=logA−logB.

Find the logarithms of each number.

Perform the addition and subtraction.

Find the antilogarithm.

Recall the formula for the volume of a cylinder.

Substitute the given values.

Solve for hhh.

Express 2187 and 243 as powers of 3.

Substitute the index forms into the equation for hhh.

Apply the division rule for indices am/an=am−na^m / a^n = a^{m-n}am/an=am−n.

Write the formula for the area of a rectangle.

Substitute the given expressions for length, width, and area.

Expand the right side of the equation.

Rearrange the equation into a standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

Solve the quadratic equation for xxx.

Choose the valid value for xxx.

Calculate the dimensions using x≈8.947x \approx 8.947x≈8.947.

Identify prime numbers in this range (numbers only divisible by 1 and themselves).

All other numbers in the list are composite.

Still stuck on this one?

More Mathematics Questions

Apply the power rule $(a^m)^n = a^{mn}$ .

Apply the product rule $a^m \times a^n = a^{m+n}$ .

Solve for $x$ .

Apply logarithm properties $\log(AB) = \log A + \log B$ and $\log(\frac{A}{B}) = \log A - \log B$ .

Solve for $h$ .

Substitute the index forms into the equation for $h$ .

Apply the division rule for indices $a^m / a^n = a^{m-n}$ .

Rearrange the equation into a standard quadratic form $ax^2 + bx + c = 0$ .

Solve the quadratic equation for $x$ .

Choose the valid value for $x$ .

Calculate the dimensions using $x \approx 8.947$ .