how to solve for c

Everly

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10-01-2025

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Solving for a variable, like 'c', means isolating it on one side of an equation. The exact method depends on the equation's complexity. This guide will walk you through various scenarios, from simple equations to more advanced ones involving exponents and logarithms.

Solving Simple Equations for 'c'

Let's start with the basics. If 'c' is involved in simple addition, subtraction, multiplication, or division, the solution is straightforward.

Example 1: c + 5 = 10

To isolate 'c', subtract 5 from both sides of the equation:

c + 5 - 5 = 10 - 5

c = 5

Example 2: c - 7 = 3

Add 7 to both sides:

c - 7 + 7 = 3 + 7

c = 10

Example 3: 3c = 12

Divide both sides by 3:

3c / 3 = 12 / 3

c = 4

Example 4: c / 4 = 6

Multiply both sides by 4:

(c / 4) * 4 = 6 * 4

c = 24

Solving More Complex Equations for 'c'

Things get a bit trickier when 'c' is involved in more complex algebraic expressions.

Example 5: 2c + 7 = 15

1. Subtract 7 from both sides: 2c = 8
2. Divide both sides by 2: c = 4

Example 6: 5c - 3 = 2c + 9

1. Subtract 2c from both sides: 3c - 3 = 9
2. Add 3 to both sides: 3c = 12
3. Divide both sides by 3: c = 4

Example 7: (c + 2)/3 = 5

1. Multiply both sides by 3: c + 2 = 15
2. Subtract 2 from both sides: c = 13

Solving Equations with Exponents for 'c'

When 'c' is an exponent, we need to use logarithms.

Example 8: 2^c = 8

1. Take the logarithm of both sides (using any base, but base 10 or e are common): log(2^c) = log(8)
2. Use the logarithm power rule (log(a^b) = b * log(a)): c * log(2) = log(8)
3. Divide both sides by log(2): c = log(8) / log(2)
   This simplifies to c = 3 (since 2³ = 8)

Example 9: e^c = 10

1. Take the natural logarithm (ln) of both sides: ln(e^c) = ln(10)
2. Since ln(e^x) = x, we have: c = ln(10) You'll need a calculator to find the approximate value of ln(10).

Solving Equations with Logarithms for 'c'

If 'c' is within a logarithm, we use the properties of logarithms to solve.

Example 10: log₁₀(c) = 2

1. Convert the logarithmic equation to exponential form: 10² = c
2. Simplify: c = 100

Example 11: ln(c) = 5

1. Convert to exponential form using base e: e⁵ = c
2. Use a calculator to find the approximate value of e⁵.

Troubleshooting and Tips

• Check your work: Substitute your solution back into the original equation to verify it's correct.
• Use a calculator: For complex calculations involving logarithms or exponents.
• Practice: The more you practice, the more comfortable you'll become with solving for 'c' in various equations.
• Break down complex equations: Simplify complex equations into smaller, manageable steps.

This comprehensive guide should equip you to solve for 'c' in a wide range of mathematical scenarios. Remember that the key is to isolate 'c' by applying inverse operations to both sides of the equation. If you encounter a particularly challenging problem, don't hesitate to break it down into smaller, more manageable parts.