Question

Solve. Check your answers using substitution. a) \(2^{4 x}=4^{x+3}\) b) \(25^{x-1}=5^{3 x}\) c) \(3^{w+1}=9^{w-1}\) d) \(36^{3 m-1}=6^{2 m+5}\)

Short answer

a) \(x=3\). b) \(x=2\). c) \(w=3\). d) \(m=2\).

Step-by-step solution

Convert bases to same base

Convert all terms so they have the same base. For example, since 4 is 2^2 and 25 is 5^2, rewrite the equations.

Rewrite equations using the same bases

a) \(2^{4 x}=(2^2)^{x+3}\) which simplifies to \(2^{4 x}=2^{2(x+3)}\). b) \(25^{x-1}=(5^2)^{x-1}\) which simplifies to \(5^{2(x-1)}=5^{3 x}\). c) \(3^{w+1}=(3^2)^{w-1}\) which simplifies to \(3^{w+1}=3^{2(w-1)}\). d) \(36^{3 m-1}=(6^2)^{3 m-1}\) which simplifies to \(6^{2(3 m-1)}=6^{2 m+5}\).

Equate the exponents

Since the bases are now the same, set the exponents of each term equal to each other. For example, for \(2^{4 x} = 2^{2(x+3)}\), we equate exponents \(4x = 2(x+3)\).

Solve for the variable

a) \(4x = 2(x+3)\) implies solving for x. b) \(2(x-1) = 3x\) implies solving for x. c) \(w+1=2(w-1)\) implies solving for w. d) \(2(3m-1) = 2m+5\) implies solving for m.

Substitution Check

Substitute the values found back into the original equation to verify the solutions.

Key concepts

solving exponential equations

To solve exponential equations, we need to make the exponents comparable. This typically involves expressing the equations with the same base.
Once the bases are the same, the corresponding exponents can be set equal to each other.
Let's walk through an example to make this clearer.
Consider the equation: \(2^{4x} = 4^{x+3}\).
Notice that 4 can be expressed as a power of 2, specifically, \(4=2^2\).
By rewriting the bases, the equation becomes \(2^{4x} = (2^2)^{x+3}\).
Simplifying the right-hand side, we get \(2^{4x} = 2^{2(x+3)}\).
Now, both sides of the equation have the same base (2).
Thus, we equate the exponents: \(4x = 2(x+3)\).
Solving this for x, we get \(4x = 2x + 6\), and \(2x = 6\), so \(x = 3\).
Always remember to check your solution by plugging it back into the original equation.

exponential functions

Exponential functions are mathematical functions in the form \(f(x) = a \times b^{x}\), where \(a\) is a constant, \(b\) is the base of the exponential, and \(x\) is the exponent.
These functions grow extremely rapidly, which is their most notable characteristic.
For example, in the function \(f(x) = 2^{x}\), the base is 2, and as \(x\) increases, the value of \(f(x)\) grows quickly.
Recognizing such functions can help you solve exponential equations more efficiently.
The equations we deal with often require converting bases or exponents, so understanding how exponential functions behave is crucial.

substitution method

The substitution method involves replacing variables with known values to simplify and solve equations.
This method can be incredibly useful for checking solutions in exponential equations.
Let's revisit the equation \(2^{4x} = 4^{x+3}\) and its solution \(x = 3\).
Substituting \(x = 3\) back into the original equation, we get: \(2^{4 \times 3} = 4^{3 + 3}\), which simplifies to \(2^{12} = 4^6\).
Rewriting 4 again as \(2^2\), we see that \(4^6 = (2^2)^6 = 2^{12}\).
Thus, both sides of the equation are equal, confirming the solution \(x = 3\).
This method not only verifies the correctness of our solution but also cements our understanding of the problem.

rewriting bases

Rewriting bases is a crucial step in solving exponential equations.
This technique allows us to transform the equation into an easier-to-solve form.
To illustrate, consider the equation \(25^{x-1} = 5^{3x}\).
Notice that 25 can be written as \(5^2\).
So, we rewrite the equation as \((5^2)^{x-1} = 5^{3x}\).
Simplifying the left side, we get \(5^{2(x-1)} = 5^{3x}\).
With the same base (5) on both sides, we equate the exponents: \(2(x-1) = 3x\).
Solving for x, we get: \(2x - 2 = 3x\), which simplifies to \(-2 = x\).
By rewriting bases, we simplify the initial complex equation, making it much easier to solve.