Solve an exponential equation by matching bases

Solve this equation for x.

An amount starts at 3 and doubles after each complete period. Change n and compare the amount with the target 48. Count periods from 0

At which whole period does the amount first equal 48?

Now solve an equation with a reciprocal target

A reciprocal power has a negative exponent

Match the base before comparing indices

For a positive base other than 1, equal powers have equal indices. A negative index means a reciprocal, not a negative value

1. 1Rewrite the target The reciprocal target is still positive \(\displaystyle \begin{aligned}\frac18&=\frac{1}{2^3}\\&=2^{-3}\end{aligned}\)
2. 2Compare indices Both sides now have base 2 \(\displaystyle 2^x=2^{-3}\quad\Longrightarrow\quad x=-3\)

The index is now an expression. Find x exactly

After matching the base, solve the resulting linear equation

Finish solving the index expression

The matched exponent is 5, but the question asks for x, not for three times x plus one

1. 1Match Rewrite 32 using base 2. \(\displaystyle 2^{3x+1}=2^5\)
2. 2Solve Subtract 1, then divide by 3. \(\displaystyle \begin{aligned}3x+1&=5\\3x&=4\\x&=\frac43\end{aligned}\)

Solve this different index expression exactly

An amount starts at 80 units at period 0 and is divided by 2 after each complete period. How many complete periods does it take to equal 5 units?

Divide the target by the initial amount, then match the reciprocal power

An amount starts at 5 units at period 0 and triples each complete period. Which is the first period when it is at least 100 units?

A first threshold needs two checks

An amount starts at 5 units at period 0 and triples each complete period. Find the first period when it is at least 100 units. Meeting the target is not enough: check that the previous whole period was still below it. An exact equality also satisfies 'at least'

1. 1Previous period The target has not yet been reached \(\displaystyle A_2=5\times3^2=45<100\)
2. 2First qualifying period All earlier amounts are smaller because the amount increases \(\displaystyle A_3=5\times3^3=135\ge100\)

A different amount starts at 2 units at period 0 and triples after each complete period. Find the first whole period when it is at least 162 units

Compare the exact amounts at a candidate period and the period immediately before it