7 If logs(x) = 2, find x. 8 Solve: 3x+1 = 81 9 A bacterial culture doubles every hour. If there are initially…

Question

7 If logs(x) = 2, find x. 8 Solve: 3x+1 = 81 9 A bacterial culture doubles every hour. If there are initially 100 bacteria, how many will there be after 5 hours? 10 The pH of a solution is given by pH = -log10[H+]. If [H+] = 10-7, what is the pH?

ScanSolve AI Tutor · Accepted Answer

Problem 7: If log₅(x) = 2, find x. (Assuming the base of the logarithm is 5, as this leads to one of the options.) Step 1: Convert the logarithmic equation to an exponential equation. log₅(x) = 2 ⇒ x = 5² Step 2: Calculate the value of x. x = 25 The correct option is D. D. x = 25; Problem 8: Solve: 3^(x+1) = 81. Step 1: Express 81 as a power of 3. 81 = 3⁴ Step 2: Equate the exponents since the bases are the same. 3^(x+1) = 3⁴ ⇒ x+1 = 4 Step 3: Solve for x. x = 4 − 1; x = 3 The correct option is B. B. x = 3 Problem 9: A bacterial culture doubles every hour. If there are initially 100 bacteria, how many will there be after 5 hours? Step 1: Identify the initial number of bacteria and the growth factor. Initial bacteria P₀ = 100. The culture doubles every hour, so the growth factor is 2. The number of hours is t = 5. Step 2: Use the exponential growth formula P(t) = P₀ · 2^t. P(5) = 100 · 2⁵ Step 3: Calculate the value. P(5) = 100 · 32; P(5) = 3200 The correct option is C. C. 3200 Problem 10: The pH of a solution is given by pH = −log₁₀[H^+]. If [H^+] = 10⁻⁷, what is the pH? Step 1: Substitute the given concentration of hydrogen ions into the pH formula. pH = −log₁₀(10⁻⁷); Step 2: Use the logarithm property log_b(b^x) = x. pH = −(−7) Step 3: Simplify the expression. pH = 7 The correct option is C. C. 7 Problem 11: Simplify: (2⁻³ × 2⁵) ÷ 2². Step 1: Apply the product rule for exponents (a^m × aⁿ = a^(m+n)) to the terms inside the parenthesis. 2⁻³ × 2⁵ = 2⁻³⁺⁵ = 2² Step 2: Substitute this result back into the expression. 2² ÷ 2² Step 3: Apply the quotient rule for exponents (a^m ÷ aⁿ = a^(m−n)). 2² ÷ 2² = 2²⁻² = 2⁰ Step 4: Simplify the expression. 2⁰ = 1 The correct option is A. A. 1 Problem 12: Find the remainder when 47 is divided by 5. Step 1: Perform the division of 47 by 5. 47 = 5 × q + r Step 2: Determine the quotient (q) and the remainder (r). 47 = 5 × 9 + 2 The remainder is 2. The correct option is B. B. 2 Problem 13: Solve: 3x ≡ 2 5. Step 1: Find the multiplicative inverse of 3 modulo 5. We need a number k such that 3k ≡ 1 5. 3 × 2 = 6 ≡ 1 5 So, the inverse of 3 modulo 5 is 2. Step 2: Multiply both sides of the congruence by the inverse (2). 2 × (3x) ≡ 2 × 2 5 6x ≡ 4 5 Step 3: Simplify the congruence. Since 6 ≡ 1 5. 1x ≡ 4 5 x ≡ 4 5 The correct option is D. D. x ≡ 4 5 Problem 14: What is 23 7? Step 1: Divide 23 by 7. 23 = 7 × q + r Step 2: Determine the quotient (q) and the remainder (r). 23 = 7 × 3 + 2 The remainder is 2. The correct option is B. B. 2 Problem 15: Solve: 2x + y = 7 and x − y = 2. Step 1: Use the elimination method by adding the two equations. (2x + y) + (x − y) = 7 + 2; 3x = 9 Step 2: Solve for x. x = 9/3; x = 3; Step 3: Substitute the value of x into the second equation (x − y = 2) to find y. 3 − y = 2; y = 3 − 2; y = 1; The solution is x=3, y=1. The correct option is A. A. x=3, y=1; Problem 16: Solve: 3x + 2y = 12 and x + y = 5. Step 1: From the second equation, express y in terms of x. y = 5 − x Step 2: Substitute this expression for y into the first equation. 3x + 2(5 − x) = 12 Step 3: Solve for x. 3x + 10 − 2x = 12; x + 10 = 12; x = 12 − 10; x = 2 Step 4: Substitute the value of x back into the expression for y. y = 5 − 2; y = 3; The solution is x=2, y=3. The correct option is A. A. x=2, y=3

7 If logs(x) = 2, find x. 8 Solve: 3x+1 = 81 9 A bacterial culture doubles every hour. If there are initially 100 bacteria, how many will there be after 5 hours? 10 The pH of a solution is given by pH = -log10[H+]. If [H+] = 10-7, what is the pH?

Answer

Step-by-step solution

Convert the logarithmic equation to an exponential equation.

Calculate the value of $x$ .

Express 81 as a power of 3.

Equate the exponents since the bases are the same.

Solve for $x$ .

Identify the initial number of bacteria and the growth factor.

Use the exponential growth formula $P(t) = P_0 \cdot 2^t$ .

Calculate the value.

Substitute the given concentration of hydrogen ions into the pH formula.

Use the logarithm property $\log_b(b^x) = x$ .

Simplify the expression.

Apply the product rule for exponents ( $a^m \times a^n = a^{m+n}$ ) to the terms inside the parenthesis.

Substitute this result back into the expression.

Apply the quotient rule for exponents ( $a^m \div a^n = a^{m-n}$ ).

Simplify the expression.

Perform the division of 47 by 5.

Determine the quotient ( $q$ ) and the remainder ( $r$ ).

Find the multiplicative inverse of 3 modulo 5.

Multiply both sides of the congruence by the inverse (2).

Simplify the congruence.

Divide 23 by 7.

Determine the quotient ( $q$ ) and the remainder ( $r$ ).

Use the elimination method by adding the two equations.

Solve for $x$ .

Substitute the value of $x$ into the second equation ( $x - y = 2$ ) to find $y$ .

From the second equation, express $y$ in terms of $x$ .

Substitute this expression for $y$ into the first equation.

Solve for $x$ .

Substitute the value of $x$ back into the expression for $y$ .

Try one yourself

Still stuck on this one?

More Mathematics Questions

7 If logs(x) = 2, find x. 8 Solve: 3x+1 = 81 9 A bacterial culture doubles every hour. If there are initially 100 bacteria, how many will there be after 5 hours? 10 The pH of a solution is given by pH = -log10[H+]. If [H+] = 10-7, what is the pH?

Answer

Step-by-step solution

Convert the logarithmic equation to an exponential equation.

Calculate the value of xxx.

Express 81 as a power of 3.

Equate the exponents since the bases are the same.

Solve for xxx.

Identify the initial number of bacteria and the growth factor.

Use the exponential growth formula P(t)=P0⋅2tP(t) = P_0 \cdot 2^tP(t)=P0​⋅2t.

Calculate the value.

Substitute the given concentration of hydrogen ions into the pH formula.

Use the logarithm property log⁡b(bx)=x\log_b(b^x) = xlogb​(bx)=x.

Simplify the expression.

Apply the product rule for exponents (am×an=am+na^m \times a^n = a^{m+n}am×an=am+n) to the terms inside the parenthesis.

Substitute this result back into the expression.

Apply the quotient rule for exponents (am÷an=am−na^m \div a^n = a^{m-n}am÷an=am−n).

Simplify the expression.

Perform the division of 47 by 5.

Determine the quotient (qqq) and the remainder (rrr).

Find the multiplicative inverse of 3 modulo 5.

Multiply both sides of the congruence by the inverse (2).

Simplify the congruence.

Divide 23 by 7.

Determine the quotient (qqq) and the remainder (rrr).

Use the elimination method by adding the two equations.

Solve for xxx.

Substitute the value of xxx into the second equation (x−y=2x - y = 2x−y=2) to find yyy.

From the second equation, express yyy in terms of xxx.

Substitute this expression for yyy into the first equation.

Solve for xxx.

Substitute the value of xxx back into the expression for yyy.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Calculate the value of $x$ .

Solve for $x$ .

Use the exponential growth formula $P(t) = P_0 \cdot 2^t$ .

Use the logarithm property $\log_b(b^x) = x$ .

Apply the product rule for exponents ( $a^m \times a^n = a^{m+n}$ ) to the terms inside the parenthesis.

Apply the quotient rule for exponents ( $a^m \div a^n = a^{m-n}$ ).

Determine the quotient ( $q$ ) and the remainder ( $r$ ).

Determine the quotient ( $q$ ) and the remainder ( $r$ ).

Solve for $x$ .

Substitute the value of $x$ into the second equation ( $x - y = 2$ ) to find $y$ .

From the second equation, express $y$ in terms of $x$ .

Substitute this expression for $y$ into the first equation.

Solve for $x$ .

Substitute the value of $x$ back into the expression for $y$ .